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Organizers |
An algebraic construction of colored HOMFLY-PT homology
by
Michael Abel
Virginia Commonwealth University
Coauthors: Matt Hogancamp
We construct complexes of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. Using these categorical idempotents, we construct a triply graded link homology theory which categorifies the HOMFLY-PT polynomial colored by one-column partitions. As an application of this theory, we compute the stable Khovanov-Rozansky homology of torus knots HHH(T(n, )). This link homology theory specializes to Khovanov-Rozansky homology in the uncolored case.
Date received: December 10, 2014
Heegaard diagrams corresponding to Turaev surfaces
by
Cody Armond
University of Iowa
Coauthors: Nathan Druivenga, Thomas Kindred
Turaev surfaces are special surfaces constructed from link diagrams which the link projects to in an alternating fashion.
We describe a correspondence between Turaev surfaces of link diagrams and special Heegaard diagrams adapted to the links.
Date received: December 1, 2014
Link Polynomial Calculus and AENV Conjecture
by
Semeon Arthamonov
Rutgers, The State University of New Jersey
Coauthors: A. Mironov, A. Morozov, And. Morozov
Recently Aganagic, Ekholm, Ng, and Vafa conjectured a relation between the augmentation variety in the large N limit of the colored HOMFLY and quantum A-polynomials. In this talk I will describe the methods used for direct confirmation of this conjecture for certain links including Whitehead link and Borromean rings.
It appears that colored knot polynomials possess an internal structure (we call it Z-expansion) which behaves naturally under inclusion of the representation into the product of the fundamental ones. In particular, for some large families of links the colored HOMFLY polynomial for symmetric and anti-symmetric representations can be presented as a truncated sum of a certain q-hypergeometric series. The latter allows us to extend the formulas for the arbitrary symmetric representations and study the asymptotic of the colored HOMFLY polynomials for large symmetric representations.
In addition I will say a few words about the extension of Z-expansion beyond the symmetric representations for some simplest examples. Although for generic representation we no longer have those cute truncated q-hypergeometric series we still have some interesting structure beyond the HOMFLY and superpolynomials. In particular, the introduction of the recently developed fourth grading in all existing examples can be presented as an elegant redefinition of the constituents of Z-expansion.
Date received: December 8, 2014
Trace of the categorified quantum groups
by
Anna Beliakova
University of Zurich
Coauthors: Zaur Guliyev, Kazuo Habiro, Aaron Lauda, and Ben Webster
In this talk I will give a gentle introduction to the categorified quantum groups and show that the trace (or 0th Hochschild homology) of the Khovanov-Lauda 2-category is isomorphic to the current algebra. Then I'll discuss some applications of this fact to link homology theories.
Date received: December 3, 2014
H-valued Knotoid Invariants
by
Alexander Borland
The Ohio State University
Coauthors: Thomas Kerler (The Ohio State University)
In 2011, Turaev introduced the monoid of Knotoids K. Given a ribbon Hopf algebra H together with a ribbon automorphism on H, we construct a morphism of monoids from K to H. In the case of a trivial automorphism, the construction specializes to known knot invariants that associate to a knot an element in the center of H. Moreover, when specialized to the fundamental representation of quantum sl2 our invariant reproduces Turaev's knotoid bracket polynomial with an additional parameter. The method readily implies colored versions of Tuarev's invariant as well as multi-parameter generalizations to higher rank quantum groups.
Date received: December 8, 2014
Heron's Formula from a 4-dimensional point of view
by
J. Scott Carter
University of South Alabama
Coauthors: David Mullens
Let a triangle with edge lengths a, b, and c, with a ≤ b ≤ c be given. Let A denote the area of the triangle.
Heron's formula states that
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Date received: December 7, 2014
Spectra for volume and determinant density
by
Abhijit Champanerkar
College of Staten Island and The Graduate Center, CUNY
Coauthors: Ilya Kofman and Jessica Purcell
We study the asymptotic behaviour of two basic knot invariants, a geometric invariant called the volume density defined as volume per crossing number, and a diagrammatic invariant called the determinant density defined as 2 pi log det(K) per crossing number. We will discuss theorems and conjectures relating the asymptotic behaviour of these invariants.
Date received: December 26, 2014
Low Dimensional Representations of the loop braid group LB3
by
Liang Chang
Texas A&M University
Coauthors: Paul Bruillard, Cesar Galindo, Seung-Moon Hong, Ian Marshall, Julia Plavnik, Eric Rowell and Michael Sun
The loop braid group LB3 is defined as the motion group of 3 unknotted and unlinked oriented circles in R3, which is a generalization of the 3-strand braid group B3.
In this talk, we will report the results on the low dimensional representations of LB3 extended from the representations of B3.
Date received: December 2, 2014
Algorithmic complexity of orders of groups
by
Jennifer Chubb
University of San Francisco
Coauthors: Mieczyslaw Dabkowski and Valentina Harizanov
A group is called computable if membership in the structure (as a set) can be effectively determined and there is an effective algorithm for computing the group operation. An ordering of the elements of a group is called a bi-ordering if it is invariant under the left and right actions of the group on itself. We consider the algorithmic complexity of bi-orderings admitted by a large class of residually nilpotent groups that includes surface groups. This work is joint with Valentina Harizanov and Mietek Dabkowski.
Date received: January 3, 2015
From quantum foundations via natural language meaning via string diagrams.
by
Bob Coecke
Oxford University
Earlier work on an entirely diagrammatic formulation of quantum theory, which is soon to appear in the form of a textbook [CK], has somewhat surprisingly guided us towards providing an answer for the following question: how do we produce the meaning of a sentence given that we understand the meaning of its words? The correspondence between these seemingly far apart areas was established in terms of string diagrams [CSC, C], and more recently, also internal Frobenius algebras and completely positive maps have started to play a key role in both areas. This work has practical applications in the area of Natural Language Processing. The fact that meaning in natural language, depending on the subject domain, encompasses discussions within any scientific discipline, we obtain a template for theories such as animal behaviour, social interaction, and many others.
[CK] B. Coecke and A. Kissinger (2014) Picturing Quantum Processes. Cambridge UP.
[CSC] B. Coecke, M. Sadrzadeh and S. Clark (2010) Mathematical foundations for a compositional distributional model of meaning. Linguistic Analysis - Lambek festschrift. arXiv:1003.4394
[C] B. Coecke (2012) The logic of quantum mechanics - Take II. arXiv:1204.3458
Date received: December 9, 2014
Reasons to love Chebyshev knots and billiard table diagrams
by
Moshe Cohen
Technion - Israel Institute of Technology
Jones and Przytycki showed that Lissajous knots are equivalent to billiard knots in a cube; however not all knots are Lissajous. Recently Koseleff and Pecker developed an analogous parametrization using Chebyshev polynomials and showed that all knots can arise in this way.
We highlight some facets of this model and use it to compute the Jones polynomials of 2- and 3-bridge knots.
Supported in part by the funding from the European Research Council under the European Unions Seventh Framework Programme, Grant FP7-ICT-318493-STREP.
Date received: December 23, 2014
Framed Cord Algebra Invariant of Knots in S1 ×S2
by
Xingshan Cui
University of California Santa Barbara
Coauthors: Zhenghan Wang
We generalize Ng's two-variable algebraic/combinatorial 0-th framed knot contact homology for framed oriented knots in S3 to knots in S1 ×S2, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng's three-variable knot contact homology. Our main tool is Lin's generalization of the Markov theorem for braids in S3 to braids in S1 ×S2. We conjecture that our framed cord algebras are always finitely generated for non-local knots.
Date received: December 4, 2014
Tangle analysis of protein-DNA complexes.
by
Isabel Darcy
University of Iowa
Just like local knots can occur in long extension cords, such knots can also appear in DNA. DNA can be be either linear or circular. Some proteins will cut DNA and change the DNA configuration before resealing the DNA. Thus, if the DNA is circular, the DNA can become knotted. Protein-DNA complexes were first mathematically modeled using tangles in Ernst and Sumners seminal paper, Ä calculus for rational tangles: applications to DNA recombination" (Math Proc Camb Phil Soc, 1990). A tangle consists of arcs properly embedded in a 3-dimensional ball. In order to model protein-bound DNA, the protein is modeled by the 3D ball while the segments of DNA bound by the protein can be thought of as arcs embedded within the protein ball. This is a very simple model of protein-DNA binding, but from this simple model, much information can be gained. The main idea is that when modeling protein-DNA reactions, one would like to know how to draw the DNA. For example, are there any crossings trapped by the protein complex? How do the DNA strands exit the complex? Is there significant bending? Tangle analysis cannot determine the exact geometry of the protein-bound DNA, but it can determine the overall entanglement of this DNA, after which other techniques may be used to more precisely determine the geometry.
Date received: December 5, 2014
Generalizing connected sums of knots, and the colored Jones polynomial
by
Oliver Dasbach
Louisiana State University
Coauthors: Mustafa Hajij (LSU)
A natural operation on oriented knots is given by the connected sum. Polynomial knot invariants like the Alexander polynomial or the Jones polynomial behave nicely under this operation. We will discuss other sums for knots, and the implications for the colored Jones polynomial.
Date received: December 12, 2014
The Homflypt skein module of L(p, 1): The braid approach.
by
Ioannis Diamantis
National Technical University of Athens
Coauthors: Sofia Lambropoulou, Jozef Przytycki
We represent links/braids in L(p, 1) by mixed links/braids in S3. The mixed braids are elements of the Artin Braid groups of type B, B1, n. We then give the analogue of Markov's Theorem in L(p, 1) in terms of braid equivalence in ∪n∞B1, n. The braid groups B1, n are represented in the generalized B-type Hecke algebras H1, n(q), on which a unique Markov trace has been constructed by Lambropoulou. We then give a new basis, Λ, for the Homflypt skein module of the solid torus, which was conjectured by Przytycki.
The new basis Λ is appropriate for computing the Homflypt skein module of the lens spaces, since the handle-slide moves are best described in terms of the basis Λ. Then, for computing the Homflypt skein module of L(p, 1) we solve an infinite system of equations resulting from the handle-slide moves. This system consists in finite self-contained subsystems.
Date received: December 1, 2014
Quasimaximal vector spaces
by
Rumen Dimitrov
Department of Mathematics, Western Illinois University, Macomb IL 61455
Coauthors: Valentina Harizanov
The cohesive sets appear to be the sets that you may encounter in a course in general topology. But the complements of some cohesive sets are computably enumerable and such complements are called maximal sets. The intersections of finitely many maximal sets are called quasimaximal. Quasimaximal sets and quasimaximal vector spaces play an important role in computability and computable model theory. We will explore some interesting combinatorial properties of the filters of the quasimaximal spaces in the lattice of computably enumerable vector spaces modulo finite dimension.
Date received: December 18, 2014
AJ-conjecture for Cables of Some Alternating Knots
by
Nathan Druivenga
The University of Iowa
The AJ-conjecture for a knot K ⊂ S3 relates the A-polynomial and the colored Jones polynomial of K. If K satisfies the AJ-conjecture, we give sufficient conditions on K for the (r, 2)-cable knot C to also satisfy the AJ-conjecture.
Date received: December 3, 2014
Twisting bordered Khovanov homology
by
Nguyen D. Duong
The University of Alabama
I will describe a bordered version of totally twisted Khovanov homology by twisting bordered Khovanov homology. Like bordered Khovanov homology, the twisted invariants of tangles come in two versions, so-called type A and type D structures. The type A and type D structures admit explicit spanning-tree-like deformation retractions.
Date received: December 8, 2014
The span of the Jones polynomial of a virtual knot
by
Heather A. Dye
McKendree University
The Kauffman-Murasugi-Thistlethwaite theorem gives a bound on the span of the Jones polynomial for
classical knots. In the early 2000's, Naoko Kamada published two papers that extend this result to alternating virtual links:
Span of the Jones polynomial of an alternating virtual link (Algebr. Geom. Topol. 4 (2004) 1083-1101) and On the Jones polynomials of checkerboard colorable virtual knots (arxiv, 2000).
We establish a bound on the Jones polynomial of knots that are not checkerboard colorable.
The proof uses cut points and includes the original result on checkerboard colorable knots as a special case. (This is work in progress.)
Date received: November 23, 2014
Monotonic simplification and contact topology
by
Ivan Dynnikov
Steklov Mathematical Institute
Coauthors: Maxim Prasolov
Rectangular (or grid) diagrams provide for a good formalism for describing ordinary links, closed braids, Legendrian and transverse links. We show that a rectangular diagram admits a simplification by elementary moves if and only if one of the two Legendrian links associated to the diagram admits a destabilization. This has a number of consequences including the generalized Jones conjecture about braids and the existence of an algorithm to determine the maximal Thurston-Bennequin number of a link. This may also have some more consequences for algorithmic classification of links if certain results about Legendrian links are proven.
Date received: December 22, 2014
Foundations of Topological quandles.
by
Mohamed Elhamdadi
University of South Florida
We give an elementary treatise of the foundations of topological quandles. Specifically, we give analogous notions of the concepts of ideals, normal subgroups, etc., in the context of topological quandles. This is a joint work in progress with E. Moutuou.
Date received: December 21, 2014
Some knots whose complements are homotopy circles
by
Greg Friedman
Texas Christian University
We will present a construction of non-trivial piecewise-linear embeddings Sn-2→ Sn, n ≥ 5, that are locally-flat except at an isolated singularity and such that the complement of the embedding is a homotopy circle. This in contrast to the classical locally-flat case in which the complement is a homotopy circle only if the embedding is the trivial (unknotted) one. The construction involves some nice applications of surgery theory and group theory.
Date received: December 1, 2014
The Localized Skein Algebra is Frobenius
by
Charles Frohman
The University of Iowa
Coauthors: Nelson Abdiel Cólon Vargas
When A in the Kauffman bracket skein relation is a primitive 2Nth root of unity, where N ≥ 3 is odd,
the Kauffman bracket skein algebra KN(F) of a finite type surface F is a ring extension of
the SL2C-characters χ(F) of the fundamental group of F. We localize by inverting the nonzero characters
to get an algebra S-1KN(F) over the function field of the character variety. We prove the algebra S-1KN(F)
is a symmetric Frobenius algebra. Along the way we prove K(F) is finitely generated, KN(F) is a finite rank module
over χ(F), and threaded primitive diagrams are units in S-1KN(F).
Date received: December 27, 2014
Theta Functions and Knots
by
Razvan Gelca
Texas Tech University
I will show how to relate theta functions and knots via representation theory, which gives an approach to abelian Chern-Simons theory different from Witten's. This is the main idea of my recent book with the same title.
Date received: November 27, 2014
Signatures and Nullities of real algebraic curves via plumbing diagrams
by
Patrick M. Gilmer
Louisiana University
Coauthors: Stepan Orevkov
A real algebraic curve is the zero set in the real projective plane of a homogenous polynomial in three variables. A non-singular real algebraic curve consists of a collection of disjoint simple smooth closed curves. Sometimes these curves acquire a semi-orientation from the complex locus of the polynomial in the complex projective plane. Sucn an orientation is called a complex orientation. We evaluate restrictions on these complex orientations arising from a generalization of the Murasugi-Tristram inequality.
Date received: November 29, 2014
Left-orderability, taut foliations, and cyclic branched covers
by
Cameron Gordon
University of Texas at Austin
It is conceivable that for a rational homology 3-sphere M, the following are equivalent: (1) pi1(M) is left-orderable, (2) M admits a co-orientable taut foliation, and (3) M is not a Heegaard Floer L-space. We will discuss these properties in the case where M is a cyclic branched cover of a knot in S3. This is joint work with Tye Lidman.
Date received: December 27, 2014
Some properties of the tail of the colored Jones polynomial.
by
Mustafa Hajij
Louisiana State University
The lowest coefficients of the colored Jones polynomial stabilize for adequate links and form a q-power series called the tail of the colored Jones polynomial.
We will discuss some properties of the tail of the colored Jones polynomial.
Date received: December 31, 2014
Strip Diagram approach to combinatorial Heegaard Floer homology
by
Carl Hammarsten
George Washington University
A 3-dimensional closed manifold Y represented by its branched spine has a canonical Heegaard decomposition. We present this decomposition graphically in the form of a Strip Diagram. We show that strip diagrams have nice properties which greatly simplify the calculation of Heegaard Floer homology. Motivated by this work, we present a combinatorial definition of a chain complex which we have proven is homotopically equivalent to the Heegaard Floer one, yet significantly smaller. Finally, we consider the presentation of a branched spine by its O-graph and show that reformulating our definition in these terms gives a clock-state type description for HF-hat of Y.
Date received: December 1, 2014
Maximal vector spaces
by
Valentina Harizanov
The George Washington University
Coauthors: Rumen Dimitrov
Maximal sets play an important role in computability theory, especially in the study of the lattice of algorithmically generated sets. They are co-atoms in its quotient lattice modulo finite sets. Similarly, maximal vector spaces play an important role in the study of the lattice of algorithmically generated vector spaces and its quotient lattice modulo finite dimension. We investigate principal filters determined by maximal spaces and how algebraic structure interacts with computability theoretic properties.
Date received: December 18, 2014
Reversion, Rotation, and Exponentiation in Dimensions Five and Six
by
Emily Herzig
University of Texas at Dallas
Coauthors: V. Ramakrishna, M. Dabkowski
An algorithm is given for computing the exponential of a skew-symmetric 5x5 or 6x6 matrix by reducing the calculations, through the conjugation and reversion maps on an appropriate Clifford algebra, to computation of the exponential of a 4x4 matrix with a known minimal polynomial.
Date received: December 30, 2014
Links with finite n-quandles
by
Jim Hoste
Pitzer College
Coauthors: Patrick Shanahan
I will discuss what is known about links with finite n-quandles and give several examples.
Date received: December 15, 2014
Left-orderability and cyclic branched coverings
by
Ying Hu
Louisiana State University
A group is called left-orderable if one can put a total order < on the set of
group elements so that inequalities are preserved by group multiplication on
the left. In this talk, we will discuss the left-orderability of fundamental groups
of cyclic branched coverings of the three sphere.
Date received: December 3, 2014
Strong and weak (1, 3) homotopies on knot projections
by
Noboru Ito
Waseda Institute for Advanced Study
Coauthors: Yusuke Takimura and Kouki Taniyama
This talk introduces topics from two recent joint works [1] and [2] about knot projections (i.e., unoriented one component generic immersed spherical curves). As is well-known, every knot projection can be related to the trivial knot projection (i.e., the simple closed curve) by a finite sequence of the first, second, and third Reidemeister moves (RI, RII, and RIII). In 2008, Hagge and Yazinski showed the non-triviality of equivalence classes of knot projections under RI and RIII by giving the first example. However, even which knot projection can be trivialized by RI and RIII is still unknown. As a first step to consider this problem, we decomposed RIII into weak and strong RIII as the way of Viro. As a result, we determined the equivalence class containing the trivial knot projection under RI and weak RIII in [1] and that of RI and strong RIII in [2]. [1] (resp. [2]) determined other classes under RI and weak RIII (resp. strong RIII).
[1] N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications 22 (2013), 1350085 (14 pages).
[2] N. Ito, Y. Takimura, and K. Taniyama, Strong and weak (1, 3) homotopies on knot projections, to appear in Osaka J. Math.
Date received: December 5, 2014
A singular braid monoid associated to knotted surfaces
by
Michal Jablonowski
University of Gdansk
We describe in this talk a view to knotted surfaces in the four space as elements of a monoid with four types of generators: two classical braid generators and two of singular braid types. We present local and global relations on words that do not change a corresponding surface-knot type.
Date received: December 20, 2014
Genus Ranges of 4-regular Rigid Vertex Graphs and Their Chord Diagrams
by
Natasha Jonoska
University of South Florida
Coauthors: Masahico Saito
A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. Four-regular rigid vertex graphs are closely related to chord diagrams (a circle with line segments, called chords, whose endpoints are attached to distinct points on the circle). These structures appear in studies of DNA recombination processes. We present results about the orientable genus ranges of these graphs and the chord diagrams. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs, and for the chord diagrams, are sets of consecutive integers. We address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. We present the intervals that can be realized as genus ranges for graphs with fixed number of vertices (chord diagrams with fixed number of chords) and we provide constructions of graphs that realize these ranges.
Date received: December 29, 2014
On the structure of the Hoste-Przytycki Link homotopy Skein Modules of oriented 3-manifolds
by
Uwe Kaiser
Boise State University
It is known that relations in the Hoste-Przytycki Link Homotopy Skein modules are inductively defined from transverse intersections of singular tori in 3-manifolds with oriented links. We will describe the resulting presentations of the skein modules. Then recent results of Stefan Friedl on centralizers of elements of 3-manifold groups and known results about centralizers in Seifert fibred manifolds can be used to describe how the structure of the skein modules depend on the prime and JSJ-decomposition of 3-manifolds. We will also review the q-homotopy skein modules defined by Przytycki from this viewpoint.
Date received: November 20, 2014
Topological Quantum Field Theory underlying quantum hyperbolic geometry.
by
Joanna Kania-Bartoszynska
National Science Foundation
Coauthors: Charles Frohman, the University of Iowa
We describe the steps needed for a construction of an extended topological quantum field theory
underlying quantum hyperbolic geometry.The description of this TQFT is skein theoretic.
The main role is played by the Kauffman bracket skein algebra at a root of unity.
Date received: December 22, 2014
Majorana Fermions, Topological Quantum Computing and the Dirac Equation
by
Louis H. Kauffman
Math UIC, 851 South Morgan Street, Chicago, Illinois 60607-7045
We discuss the relationship between Fermions and Majorana Fermions via their fusion and
operator algebras: how Majorana fermions are related to the Fibonacci model of topological
quantum computation via their fusion algebra (work with Sam Lomonaco) and how they are
related to partial topological computation via their operator algebra (Clifford algebra that
generates braid group representations). We discuss how the operator algebra for Fermions
is related to nilpotent solutions to the Dirac equation and speculate about the relationship
of this structure to quantum computing.
Date received: November 26, 2014
New Bounds on Virtual Bridge Number and Virtual Ascending Number
by
Noureen Khan
University of North Texas at Dallas
The virtual bridge number, vb(K) is the minimum number of bridges over all the Gauss diagrams realizing a virtual knot K. We review and analyze the notion to generalize some other virtual knot theory invariants, in particular, the virtual ascending number, av(K). We show that there are infinitely many homotopy classes of virtual knots each of which contains vb(K) = av(K). Some fundamental results and a table of invariants, vb(K) and av(K) for virtual knots with real crossing number less than 5 are given.
Date received: December 1, 2014
Hopf algebras and their categorifications via planar diagrammatics
by
Mikhail Khovanov
Columbia University
We will review how various families of Hopf algebras, including quantum groups and Hopf algebras of noncommutative symmetric functions, can be defined via bilinear forms of diagrammatical origin. Related diagrammatical calculi are crucial for categorifications of these Hopf algebras.
Date received: December 30, 2014
Geometrically and diagrammatically maximal knots
by
Ilya Kofman
College of Staten Island and The Graduate Center, CUNY
Coauthors: Abhijit Champanerkar, Jessica Purcell
The ratio of volume to crossing number of a hyperbolic knot is bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We show that many families of alternating knots and links simultaneously maximize both ratios.
Date received: December 29, 2014
On the framization of knot algebras
by
Sofia Lambropoulou
National Technical University of Athens, Greece
Coauthors: Jesus Juyumaya, Univ. of Valparaiso, Chile
We presents results on the framization of some knot algebras, defined by the speaker and Juyumaya. We explain the motivations of the concept of framization, coming from the Yokonuma-Hecke algebras, as well as recent results on the framization of the Temperley-Lieb algebra. Finally, we propose framizations for other knot algebras such as the BMW algebra, the B-type related Hecke algebras and the singular Hecke algebra.
Date received: December 2, 2014
On skein modules at roots of unity
by
Thang Le
Geogia Institute of Technology
The Kauffman bracket skein modules (KBSM) of surfaces (or 3-manifolds) are important objects in low-dimensional topology.
KBSMs have connections to knot and 3-manifold invariants, TQFT, character varieties, quantum Teichmuller spaces etc.
I will discuss the center of the skein algebras at roots of unity and their positive bases.
Date received: November 28, 2014
Twisted multi-distributivity and Lawrence representations of braid groups
by
Victoria Lebed
University of Nantes
Long and Moody suggested an ingenious procedure for transforming a representation of the braid group Bn+1 into a more sophisticated representation of Bn. For instance, their machinery upgrades the trivial representation into the Burau representation, which in turn yields the Lawrence-Krammer representation. Topological and algebraic versions of that procedure were proposed, the latter further developed by Bigelow and Tian. In this talk we will present a simple combinatorial avatar of Bigelow-Tian constructions, using double-layer colorings by what we call Alexander G-quandles. These latter originate from Ishii-Iwakiri-Jang-Oshiro's work on G-families of quandles.
Date received: December 2, 2014
The Jones polynomial, 3-braids, and L-space knots.
by
Christine R. S. Lee
Michigan State University
Coauthors: Faramarz Vafaee
A knot K in S3 is an L-space knot if K or its mirror image admits a positive L-space surgery. We will discuss how the constraints on the Alexander polynomial of such knots can be used to classify L-space knots with 3-braid representation using the Jones polynomial.
Date received: December 17, 2014
Quantum Knots and Topological Quantum Computing
by
Samuel J. Lomonaco
University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250
We will discuss how quantum knots can be used in topological quantum computing. (This is joint work wit Lou Kauffman.)
Date received: November 30, 2014
Alternating distances of knots
by
Adam Lowrance
Vassar College
An alternating distance is a knot invariant that measures how far away a knot is from alternating. Examples include
dealternating number, alternation number, Turaev genus, and alternating genus. In this talk, we give examples of families
of knots where the difference between two of the above alternating distances becomes arbitrarily large. The proofs often
use Khovanov or knot Floer homology.
Date received: December 1, 2014
Quantum computing and second quantization
by
Hanna Makaruk
Aplied Modern Physics Group, Los Alamos National Laboratory, Los Alamos, NM
Quantum computers are by their nature many particle quantum systems. Both the many‐particle
arrangement and being quantum are necessary for the existence of the entangled states, which are
responsible for the parallelism of the quantum computers. Second quantization is a very important
approximate method of describing such systems. This lecture will present the general idea of the second
quantization, and discuss shortly some of the most important formulations of second quantization.
Date received: December 5, 2014
Lower bounds for ropelength of links via higher linking numbers and other finite type invariants.
by
Andreas Michaelides
Tulane University
Coauthors: Rafal Komendarczyk
Based on the arrow diagram formulas for finite type invariants due to Goussarov, Polyak and Viro, we obtain lower bounds for the ropelength of links and knots. In this talk, I will specifically present ropelength and crossing number bounds for Brunnian links, in terms of Milnor triple linking number. Further, I will sketch how to obtain generalizations to the n-component case
Date received: December 2, 2014
HOMFLYPT skein module of the lens spaces L(p, 1)
by
Maciej Mroczkowski
University of Gdansk
Coauthors: Bostjan Gabrovsek (University of Ljubljana)
We present a computation of the HOMFLYPT skein module of lens spaces L(p, 1). It is based on link diagrams in Seifert manifolds. We show that these modules are free and present explicit bases for them.
Date received: December 22, 2014
Ribbon Biquandles and Virtual Knotted Surfaces
by
Sam Nelson
Claremont McKenna College
Coauthors: Patricia Rivera (Claremont High)
Ribbon biquandles are a type of biquandle which can be used to define an invariant of knotted and virtual knotted surfaces represented by ch-diagrams (also known as marked vertex diagrams). In this talk we will introduce the new invariant and see some examples.
Date received: November 21, 2014
Single-stranded DNA topology in Eukaryotes
by
Zbyszek Otwinowski
UT Southestern Medical Center
Coauthors: Dominika Borek
In a single chromosome, both template strands are wrapped around each other and become replicated at hundreds of
sites in parallel to form new dsDNA molecules, which are subsequently individualized into sister chromatids in prophase and
segregated to daughter cells later in anaphase. How the entanglement is avoided during individualization remains unclear. The
replication of holocentric and ring chromosomes provides much insight into this process. The products of replication do not
become entangled during mitosis for either chromosome type, and this indicates that during replication the template strands are
differentiated in a coordinated manner. Additionally, patterns of labeling in diplochromosomes that are created during
endoreduplication show that this mechanism relies on the memory of DNA strands formation, i.e. information about the order in
which two strands have been replicated is propagated across generations. Because information that differentiates strands in
dsDNA, is passed on from one generation to the next one, the mechanism of this process is by definition epigenetic.
We propose a mechanism of coordinated strand recognition that relies on formation of single-stranded topological structures,
which are generated during DNA synthesis. Structures of such type, called hemicatenanes, have been observed in plasmids,
viruses and Crenarchaeota the prokaryotes with the replication mechanism most closely related to that of eukaryotes.
Topological structures based on single-stranded DNA are likely to be a missing element that hinders our understanding of: (1)
the differentiation of the sister chromatids during their individualization, (2) the formation of higher-order structures in mitotic
chromosomes coupled to individualization, (3) ORI definition, (4) cis-regulation in epigenetic processes and (5) asymmetric cell
division, (6) spatial, temporal, and genomic co-linearity of transcriptional programs that control developmental processes.
Date received: December 15, 2014
Quantum description of knotted and linked vortices in superfluid helium
by
Robert Owczarek
University of New Mexico
Knotted and linked vortices in superfluid helium are an experimental fact. Even before it has been known that they exist I proposed to include them in the theory of the phase transition in superfluid helium. Some considerations on quantum theory of vortices in superfluid helium will be presented, possible relationship with recent developments in knot theory will be stressed.
Date received: December 2, 2014
The History of Knot Theory
by
Kenneth A. Perko, Jr.
325 Old Army Road, Scarsdale, New York 10583
Riemann's unreal surfaces begat covering spaces, which Heegaard illustrated in 1898 for functions of two complex variables. Wirtinger chopped them up into bite-sized pieces in 1905, exposing the structure of a knot's complement [cf. Epple, Branch points of algebraic functions and the beginnings of modern knot theory, Historia Mathematica 22 (1995), 384] and that, Gordon and Luecke proved in 1989, explains everything (unlike all those proliferating polynomials). Examples of historically important non-cyclic covering spaces will be exhibited, along with new, improved diagrams of an eponymous pair of knots.
Date received: December 14, 2014
Invariants of A-polynomials
by
Kate Petersen
Florida State
I'll discuss invariants of A-polynomials of knots such as degree, genus and gonality, as well as higher dimensional generalizations.
Date received: December 24, 2014
Network Based Approaches to Complex Diseases
by
Teresa Przytycka
NCBI/NIH
Complex diseases are caused by a combination of genetic and environmental factors.
In recent years network based approaches emerged as powerful tools for studying complex diseases. These built on the knowledge of physical or functional interactions between molecules I will discuss method to identify pathways dysregulated in cancer that we have developed in recent years.
Date received: December 15, 2014
Forty years in Mathematics: personal perspective
by
Jozef H. Przytycki
George Washington University
It is a perfect occasion to give some reminiscences of my 40 years of a work as a mathematician.
Date received: December 8, 2014
Degenerate distributive complex is degenerate
by
Krzysztof K. Putyra
ETH Institute for Theoretical Studies
Coauthors: Jozef H. Przytycki
We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle Q, the degenerate homology of Q is completely determined by the quandle homology of Q. For this case (and generally for two term homology of a spindle) we provide an explicit Künneth-type formula for the degenerate part. This solves the mystery in algebraic knot theory of the meaning of the degenerate quandle homology, brought over 15 years ago when the homology theories were defined, and the degenerate part was observed to be non-trivial.
Reference: arXiv:1411.5905
Date received: December 3, 2014
Generalized torsion in knot groups
by
Dale Rolfsen
University of British Columbia
Coauthors: Geoff Naylor
In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot 52 and algebraic knots in the sense of Milnor. We also identify many knots whose groups do not have generalized torsion elements.
Date received: November 30, 2014
Cocykle invariants of codimension 2 embeddings of manifolds
by
Witold Rosicki
University of Gdansk
Coauthors: J.H.Przytycki
We consider the classical problem of a position of n-dimensional manifold Mn in \Kr R n+2.
We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of knotting Mn → \Kr R n+2. In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring of a diagram of Mn embedded in \Kr R n+2 we have (n+1)- and (n+2)-(co)cycle invariants (i.e.invariant under Roseman moves).
The case n=2 is well known. The case n=3 we can explane in a geometric way. The general case we described in arXiv:1310.3030v1 .
Date received: December 8, 2014
Representations of loop braid groups
by
Eric Rowell
Texas A&M University
Coauthors: Zhenghan Wang, Zoltan Kadar, Paul Martin
Loop braid groups are the motion groups of oriented unlinks in a 3-ball. They could potentially arise as symmetries of 3-dimensional topological phases of matter admitting string-like excitations (e.g. vortices). I will report on recent work with Kadar, Martin and Wang in which we study local representations of the loop braid groups, via extensions of Yang-Baxter operators.
Date received: December 1, 2014
Group-quandle homology
by
Masahico Saito
University of South Florida
Coauthors: J.S. Carter, A. Ishii, K. Tanaka
Quandle homology theories have been constructed in analogy to group homology, and applied to classical knots and knotted surfaces. There are quandles in which group operations are partially defined, such that the two operations satisfy certain compatibility conditions. In this talk, such structures are presented, and their homology theory is defined that uses both group and quandle operations. A degenerate subcomplex is defined for triangulations of prisms, and cocycle invariants are defined from this homology theory for handle-body links.
Date received: December 13, 2014
Torsion in Khovanov homology of semi-adequate links
by
Radmila Sazdanovic
North Carolina State University
Coauthors: Jozef H. Przytycki
We analyze torsion in Khovanov homology using the correspondence between Khovanov link homology and well developed theory of Hochschild homology for algebras via chromatic graph homology. We extend A. Shumakovitchs conjecture about the existence of Z2-torsion in Khovanov link homology to semi-adequate links, and conjecture existence of higher 2-torsion.
Date received: January 1, 2015
Efficient synthesis of quantum circuits by number-theoretic methods
by
Peter Selinger
Dalhousie University
Recall that a set of quantum gates is called universal if it generates a dense subgroup of the group of all unitary operators, or in other words, if every unitary operator can be approximated up to arbitrary epsilon by a quantum circuit built from the gate set. The problem of finding such approximations is sometimes called the approximate synthesis problem. Until recently, the state-of-the-art solution to this problem was the Solovay-Kitaev algorithm, which yields circuits of size O(log^c(1/epsilon)), where c > 3. I will talk about a new efficient class of synthesis algorithms that were developed in the last two years and that are based on algebraic number theory. These algorithms achieve circuit sizes of O(log(1/epsilon)), which is also a lower bound. For some gate sets, such as the Clifford+T set or the Pauli+V set, the algorithms are strictly optimal (in absolute terms, i.e., not just up to big-O notation). For other gate sets, such as the Fibonacci anyon gate set, the algorithms are only asymptotically optimal. To round off the presentation, I will also give a presentation of the group of single-qubit Fibonacci circuits in terms of generators and relations, also based on algebraic number theory. Based on joint work with Neil J. Ross and Travis Russell.
Date received: December 30, 2014
About character varieties which are not spaces of characters.
by
Adam S. Sikora
University at Buffalo, SUNY
The coordinate rings of SL_n-character varieties are generated by trace functions. We will discuss character varieties whose coordinate rings are not generated by trace functions. (This subject relates to low-dimensional topology through quantum topology and other ways.)
Date received: December 31, 2014
On a conjecture by Kauffman on alternative and pseudoalternating links
by
Marithania SilveroCasanova
Universidad de Sevilla
In 1983 Louis Kauffman introduced the family of alternative links, as a generalization of alternating links. It is known that alternative links are pseudoalternating. Kauffman conjectured the converse.
In this talk we show that both families are equal in the particular case of knots of genus one. However, Kauffman's Conjecture does not hold in general, as we also show by finding two counterexamples. In the way we will deal with the intermediate family of homogeneous links, introduced by Peter Cromwell; the techniques used here allow us to give an explicit characterization of homogeneous links of genus 1.
Date received: December 1, 2014
Minimal generating sets of Reidemeister moves
by
Piotr Suwara
University of Warsaw
Considering Reidemeister moves with orientation of strands, one obtains 16 different Reidemeister moves. It has been shown (Polyak 2010) that among these there is a set of 4 moves that generate all of them and the set is minimal.
One may consider the moves as directed - "forward" and "backward", obtaining 32 types of moves. For instance, "forward" Reidemeister moves of type I and II will be the ones that increase the number of crossings. The problem of finding a minimal generating set of these arises.
In the talk I will present the tools used to obtain Polyak's result and their application to the problem mentioned, how the new problem motivates the study of diagram invariants that are invariant under Reidemeister moves of type I and II, and some possible ways of constructing such diagram invariants.
Date received: December 2, 2014
Site-specific Gordian distances of spatial graphs
by
Kouki Taniyama
Waseda University
For two spatial embeddings of an abstract graph we define site-specific Gordian distance between them. It is defined to be the minimal number of crossing changes between two specified abstract edges. We determine site-specific Gordian distances between some pairs of spatial embeddings of some abstract graphs. It has an application to puzzle ring problem. The site-specific Gordian distance between a Milnor link and a trivial link is determined. We use covering space arguments for the proofs.
Date received: December 3, 2014
Some remarks on Burau representation in dimension 4.
by
Pawel Traczyk
University of Warsaw, Poland
Coauthors: Anzor Beridze (Shota Rutasveli State University, Batumi, Georgia)
We discuss some observed regularities for Burau matrices of those four string braids that are in the kernel of the B(4) -> B(3) homomorphism.
Date received: December 27, 2014
Some conjectures about the colored Jones polynomial
by
Anh T. Tran
University of Texas at Dallas
Coauthors: partly joint with Thang T.Q. Le and with Effie Kalfagianni.
I will discuss some conjectures about the colored Jones polynomial.
Date received: December 12, 2014
on simple ribbon knots
by
Tatsuya Tsukamoto
Osaka Institute of Technology
Coauthors: Kengo Kishimoto, Tetsuo Shibuya
An m-ribbon fusion on a link L is an m-fusion of L and an m-component trivial link O which is split from L and each of whose components is attatched by a unique band to a component of L. Note that any ribbon knot can be obtained from the trivial knot by a ribbon fusion.
The m-ribbon fusion is called a simple ribbon fusion if O bounds m mutually disjoint disks D which are split from L such that each disk of D intersects with one of the bands B for the ribbon fusion exactly once at a single arc of ribbon type and each band of B intersects with one disk of D exactly once.
We call a knot obtained from the trivial knot by a finite sequence of simple ribbon fusions a simple ribbon knot. All ribbon knots with no more than 9 crossings, Kinoshita-Terasaka knot, and Kanenobu knots are simple ribbon knots. In this talk we give a necessary condition for satellite knots to be simple ribbon knots. As a consequence, we show that a (p,q)-cable of any ribbon knot (p>1) is not a simple ribbon knot.
Date received: December 12, 2014
Pictural calculus of isometries
by
Oleg Viro
Stony Brook University
In many homogeneous spaces any isometry is a composition of two involutions which are defined by
their fixed point sets. Hence, an isometry is presented by an ordered pair of subspaces, the
fixed point sets of the involutions. In low-dimensional spaces compositions of isometries can be
easily expressed in terms this presentation. We will discuss multiplication rules similar to
the head to tail addition of vectors for all isometries of plane, 3-space, 2-sphere, projective
plane, hyperbolic plane. This will be compared to Hamilton's presentation of quaternions as
fractions of vectors.
Date received: December 20, 2014
A Khovanov Approach to Quiver Cohomology
by
Jing Wang
George Washington University
Coauthors: Jozef H. Przytycki
In this talk I will give a Khovanov-type definition of quiver cohomology using the language of homology of small categories. Our work is motivated from Helme-Guizon and Rong's chromatic graph cohomology. We generalize their theory for non-commutative algebras. In particular, we consider the idea of multipaths first raised by Turner and Wagner. We discuss the relation between our construction of quiver cohomology and their work on homology of directed graphs.
Date received: December 1, 2014
Random Walk Invariants of String Links from R-Matrices
by
Yilong Wang
The Ohio State University
Coauthors: Thomas Kerler
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.
Date received: December 9, 2014
Classification of (2+1)-TQFTs and its applications to physics and quantum computation
by
Zhenghan Wang
Microsoft Station Q/UCSB
Coauthors: P. Bruillard, S.-H. Ng, E. Rowell
We classify (2+1)-TQFTs, not necessarily fully extended, by classifying their modular categories.
A recent proof of the rank-finiteness conjecture for modular categories by P. Bruillard, S.-H. Ng, E. Rowell and myself implies
that in principle we can list all modular categories of a given rank.
This classification program is inspired by the classification of 2D topological phases of matter, and is part of the mathematical
foundation of topological quantum computation.
Date received: December 15, 2014
A Khovanov-Rozansky-ish homology over Q[a]
by
Hao Wu
George Washington University
I'll show that, after a simple grading shift, the n-th transverse Khovanov-Rozansky homology stabilizes after iterated stabilization of the transverse link. The resulting stable homology is an invariant for smooth links in the form of a triply graded Q[a]-module. The sl(n) Khovanov-Rozansky homology can be easily recovered from this stable homology.
Date received: December 2, 2014
Torsion of a finite quasigroup quandle is annihilated by its order
by
Seung Yeop Yang
The George Washington University
Coauthors: Jozef H. Przytycki
We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its homology. It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. From the very beginning of the rack homology (between 1990 and 1995) the analogous result was suspected. The first general results in this direction were obtained independently about 2001 by R.A.Litherland and S.Nelson, and P.Etingof and M.Grana. In Litherland-Nelson's paper it is proven that if (Q;*) is a finite homogeneous rack (this includes quasigroup racks) then the torsion of homology is annihilated by |Q|^n. In Etingof-Grana paper it is proven that if (X;A) is a finite rack and N=|G^0_Q| is the order of a group of inner automorphisms of Q, then only primes which can appear in the torsion of homology are those dividing N (the case of connected Alexander quandles was proven before by T.Mochizuki). The result of Litherland-Nelson is generalized by Niebrzydowski and Przytycki and in particular, they prove that the torsion part of the homology of the dihedral quandle R_3 is annihilated by 3. In Niebrzydowski-Przytycki paper it is conjectured that for a finite quasigroup quandle, torsion of its homology is annihilated by the order of the quandle. The conjecture is proved by T.Nosaka for finite Alexander quasigroup quandles. In this paper we prove the conjecture in full generality. (Appeared on arXiv:1411.1727 [math.GT])
Date received: November 29, 2014
Search for Majorana Fermions in Spin-Orbit Coupled Superfluids and Superconductors
by
Chuanwei Zhang
The University of Texas at Dallas
Topological quantum matter has been an active research field in physics in the past three decades with numerous celebrated examples, including quantum Hall effect, chiral superconductor, topological insulator, etc. In topological materials, Majorana fermions, first envisioned by E. Majorana in 1935 to describe neutrinos, often emerge as topological quasiparticle excitations of the systems. Majorana fermions are intriguing because they are their own anti-particles and follow non-Abelian anyonic statistics. Although the emergence of Majorana fermions in any condensed matter or atomic system is by itself an extraordinary phenomenon, they have also come under a great deal of recent attention due to their potential use in fault tolerant quantum computation. In this talk, I will discuss recent theoretical and experimental progress on the search for Majorana fermions in two spin-orbit coupled systems: spin-orbit coupled degenerate Fermi gases and semiconductor/superconductor nanostructures. I will discuss the contribution of my group in this rapidly developing field.
Date received: December 18, 2014
Bridge number and diagrams of surface links in the 4-sphere
by
Alex Zupan
UT Austin
Coauthors: David Gay, Jeffrey Meier
Adapting work of Gay and Kirby on 4-manifolds, we introduce the notion of a bridge trisection for a surface link, which may be viewed as an analogue of a classical bridge decomposition for a link in the 3-sphere. A bridge trisection yields a presentation of a surface link as a triple of planar tangle diagrams. We discuss evidence that these triples can be used to convert classical knot invariants to dimension 4.
Date received: December 12, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.