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Organizers |
On the resolution of the wave particle duality paradox in quantum mechanics
by
Areski Nait Abdallah
Univ. of Western Ontario, Canada and INRIA Rocquencourt, France
We consider the wave particle duality paradox in quantum mechanics, and show that it appears in the form of the distinct logical fallacies. We introduce a logic of partial information framework where the paradox can be solved, and its quantum physical significance displayed.
Date received: April 3, 2016
HOMFLY-PT homology of general link diagrams up to braidlike isotopy and its decategorification
by
Michael Abel
Duke University
In the construction of HOMFLY-PT homology, one must start with a link presented as a braid closure. This restriction was expected by Khovanov and Rozansky to be required for the homology to be an isotopy invariant. In this talk we explore the consequences of dropping this requirement and allowing general link diagrams. We explicitly show that the Reidemeister IIb move (where the strands have opposite orientations) fails, and discuss the effect on defining a virtual link invariant. Finally we will show that the Euler characteristic of this homology theory is a deformed version of the HOMFLY-PT polynomial which detects "braidlike" isotopy of tangles and links. This new polynomial agrees with the HOMFLY-PT polynomial on link diagrams which are presented as closed braid diagrams.
Date received: March 31, 2016
Gauss-Gassner Invariants
by
Dror Bar-Natan
University of Toronto
In a "degree d Gauss diagram formula" one produces a number by summing over all
possibilities of paying very close attention to d crossings in some n-crossing
knot diagram while observing the rest of the diagram only very loosely, minding
only its skeleton. The result is always poly-time computable as only binom(n,d)
states need to be considered. An under-explained paper by Goussarov, Polyak, and
Viro [GPV] shows that every type d knot invariant has a formula of this kind. Yet
only finitely many integer invariants can be computed in this manner within any
specific polynomial time bound.
I suggest to do the same as [GPV], except replacing "the skeleton" with "the
Gassner invariant", which is still poly-time. One poly-time invariant that arises
in this way is the Alexander polynomial (in itself it is infinitely many
numerical invariants) and I believe (and have evidence to support my belief) that
there are more.
More at http://www.math.toronto.edu/~drorbn/Talks/NCSU-1604/.
Date received: April 17, 2016
Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group
by
Juanita Pinzon Caicedo
The University of Georgia
In the 1980s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of special homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. Hedden and Kirk then used the aforementioned criterion to establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group.
In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that simplify and extend their argument.
Date received: February 10, 2016
On twisted cocycle invariants with coefficient Z
by
Zhiyun Cheng
Beijing Normal University & George Washington University
The twisted quandle cocycle invariants of knots were introduced by J S Carter, M Elhamdadi and M Saito in 2002. In this talk I will show that if we work with coefficient Z and T=1 or -1, then the twisted cocycle invariants are trivial for knots.
Date received: March 19, 2016
A movie on knotted surfaces in 4-space
by
Ester Dalvit
University of Toronto
I will show a computer graphics movie about knotted surfaces in 4-dimensional space.
The purpose of the movie is twofold: to explain topics in knot theory in an informal and charming way to non-experts, and to provide images and animations to experts for teaching and popularization.
Contents: An introduction to knots (2d knot diagrams, Reidemeister moves, 3-colorability); The corresponding theory for knotted spheres in 4-space (3d surface diagrams, Roseman moves, 3-colorability); Ribbon 2-knots, welded knots, Satoh's Tube map relating the two, a conjecture about a Reidemeister calculus for ribbon knotted surfaces.
The movie is still work in progress. Drafts can be found at http://katlas.math.toronto.edu/ester/movies/
Date received: April 25, 2016
Ribbonlength of Knot Diagrams
by
Elizabeth Denne
Washington & Lee University
Coauthors: John M. Sullivan, Nancy Wrinkle
The ropelength problem asks to minimize the length of a knotted space curve such that a unit tube around the curve remains embedded. A two-dimensional analog has a much more combinatorial flavor: we require a unit-width ribbon around a knot diagram to be immersed with consistent crossing information. The ribbonlength is the length of the knot diagram divided by the width. In this talk I will introduce all these ideas, including a new result about the medial axis of an immersed disk in the plane.
Date received: February 24, 2016
An Invariant for Virtual Singular Links
by
Kelsey Renee Friesen
California State University, Fresno
A singular link is an immersion of a disjoint union of circles into three-dimensional space that admits only finitely many singularities that are all transverse double points. A singular link diagram is a projection of a singular link into a plane, and contains two types of crossings, namely classical crossings and singular crossings.
Virtual knot theory, introduced by Lou Kauffman in 1996, can be regarded as a projection of classical knot theory in thickened surfaces. We take one step further by studying virtual singular links, which can be thought as immersions of disjoint unions of circles into thickened surfaces. A virtual singular link diagram contains then three types of crossings: classical, singular, and virtual. Much as in the case of classical and virtual knot theory, when studying virtual singular links we seek for ways to tell them apart. An invariant for a virtual singular link is an object associated to it, which is independent on the link diagram, and may provide a powerful tool at distinguishing virtual singular links.
In this research, we employ a certain model for the sl(n) polynomial of classical links and extend it to a polynomial invariant for virtual singular links, which is defined as a state-sum formula. After this extension, we investigate how the polynomial behaves with respect to mirror images, disjoint unions, and compositions of virtual singular links.
Date received: February 15, 2016
On the Kauffman bracket skein module of the 3-torus
by
Patrick Gilmer
Louisiana State University
Recently Carrega has shown that the Kauffman bracket skein module of the 3-torus (over the field of rational functions in A) can be generated by 9 skein elements. We show this set of generators is linearly independent.
Date received: April 11, 2016
Homology of Jucys-Murphy elements and the flag Hilbert scheme
by
Eugene Gorsky
UC Davis
Coauthors: Andrei Negut, Jacob Rasmussen
Jucys-Murphy elements are known to generate a maximal commutative subalgebra in the Hecke algebra. They can be categorified to a family of commuting complexes of Soergel bimodules. I will describe a relation between a category generated by these complexes and the category of sheaves on the flag Hilbert scheme of points on the plane, using the recent work of Elias and Hogancamp on categorical diagonalization.
Date received: February 23, 2016
Tangles and the Alexander polynomial: studying related invariants
by
Iva Halacheva
University of Toronto
The Multivariable Alexander Polynomial, or MVA, is originally defined by Torres as an invariant of links. In her thesis, Archibald constructs an invariant of virtual tangles which generalizes the MVA and provides an easy verification of almost all its skein relations. I will define a reduced version of this invariant and discuss some of its properties, relations to the Gassner representation on braids and to an Alexander-type invariant on pure tangles (without closed components) introduced by Bar-Natan and coming from ribbon-knotted circles and spheres in four dimensions.
Date received: April 18, 2016
Properties of the degree colored Jones polynomials
by
Effie Kalfagianni
Michigan State University
We survey results and conjectures about topological properties and quantities of knot complements detected by the degree of colored Jones polynomials.
Date received: April 19, 2016
Elements of Khovanov Homology and Khovanov Homotopy
by
Louis H. Kauffman
Math, UIC, 851 South Morgan Street, Chicago, IL 6070-7045
This is both a research talk and an introductory talk about Khovanov homology.
We start with the bracket polynomial model of the Jones polynomial and discuss how
Khovanov homology is built from the states of the bracket polynomial by regarding
them as generating a small category. We discuss Bar-Natans tangle-cobordism
picture of Khovanov homology, and show how his 4Tu-Relation is obtained naturally
in the attempt to make the theory invariant under chain homotopy.
We show how the associated Frobenius algebras arise naturally and how all of this fits
together. We then examine the question of chain homotopy versus homotopy
and show how a judicious use of simplicial theory (the Dold-Kan Theorem)
produces natural spaces that behave well stably to give
a homotopy theory for Khovanov homology.
Date received: April 19, 2016
Reflections on categorification
by
Mikhail Khovanov
Columbia University
We will discuss various facets of categorification and its place in low-dimensional topology.
Date received: February 21, 2016
Legendrian Link Invariants
by
Caitlin Leverson
Duke University
Given a plane field dz-xdy in R3. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in R3, J1(S1), and #k(S1×S2). In particular, we will look at the relationship between two Legendrian knot invariants, normal rulings and augmentations of the Chekanov-Eliashberg differential graded algebra. No knowledge of the Chekanov-Eliashberg DGA will be assumed.
Date received: February 17, 2016
Some remarks on the nugatory crossing conjecture
by
Tye Lidman
IAS
Coauthors: Allison Moore
The nugatory crossing conjecture states that changing a non-nugatory crossing in a knot diagram changes the isotopy type of the knot. Using some basic three-manifold topology, we will give some new families of knots satisfying this conjecture.
Date received: April 2, 2016
On The Braid Index of Alternating Braids
by
Pengyu Liu
UNC Charlotte
Coauthors: Yuanan Diao and Gabor Hetyei
We define that a braid diagram is alternating if its closure is an alternating link diagram and an oriented link L is an alternating braid on n strands, if L can be represented by a reduced alternating braid diagram on n strands. In this talk, we show that the braid index of this class of alternating links, alternating braids on n strands, is exactly n.
Date received: March 29, 2016
The Jones polynomial of almost alternating and Turaev genus one links
by
Adam Lowrance
Vassar College
Coauthors: Oliver Dasbach (Louisiana State University)
We show that either the leading or trailing coefficient of the Jones polynomial of an almost alternating link (or a link of Turaev genus one) has absolute value one.
Date received: April 8, 2016
Bikei Homology
by
Sam Nelson
Claremont McKenna College
Coauthors: Jake Rosenfield
We introduce a modified homology and cohomology theory for involutory biquandles (also known as bikei). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in R 4 .
Date received: March 22, 2016
Knot contact homology and string topology
by
Lenny Ng
Duke University
Coauthors: Kai Cieliebak, Tobias Ekholm, Janko Latschev
A natural question about knot contact homology, a knot invariant with origins in contact geometry, is what information it contains about the topology of a knot. Until recently we had only a rather minimal understanding of this. I will discuss a way to describe a key part of knot contact homology, the "cord algebra", through string topology. This allows us to interpret the cord algebra in terms of the fundamental group of the knot complement, and in particular to conclude that knot contact homology detects the unknot and (by work of Gordon and Lidman) torus knots.
Date received: April 8, 2016
The augmentation category map induced by exact Lagrangian cobordisms
by
Yu Pan
Duke University
To a Legendrian knot, one can associate an A∞category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms. The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.
Date received: February 21, 2016
On Kauffman bracket skein algebras of marked surfaces and the Chebyshev-Frobenius homomorphism
by
Jonathan Paprocki
Georgia Tech
Coauthors: Thang T. Q. Le
Kauffman bracket skein algebras of marked surfaces admit both knots and arcs ending at marked points. Muller showed that skein algebras of marked surfaces embed into a very simple algebra called a quantum torus.
We show that the Chebyshev homomorphism of Bonahon and Wong between unmarked surface skein algebras at certain roots of unity is induced by a sort of Frobenius homomorphism between quantum torii obtained by marking the surface. We use this technique to extend their result to define a "Chebyshev-Frobenius" linear transformation between skein modules of (un)marked 3-manifolds at certain roots of unity.
In addition, we extend Muller's result to allow marked surfaces with unmarked boundary components. This allows us to define a surgery theory, relating skein algebras between surfaces obtained by plugging holes of boundary components and adding/removing marked points.
Date received: March 7, 2016
Branched Covers of Tricolored Knots
by
Ken Perko
325 Old Army Road, Scarsdale, New York
A brief discussion of Fox 3-colored covering spaces of knots, with diagrams of 150 examples, and a handful of additions to the list of positive Perko knots at page 167 of Kauffman et al.'s "Introductory Lectures on Knot Theory" (World Scientific, 2012): 12-n594, 12-n638, 12-n640, 12-n644 and 12-n647.
Date received: April 15, 2016
Does Khovanov homology lead to wedges of spheres?
by
Jozef H. Przytycki
George Washington University
Coauthors: Marithania Silvero (University of Zaragoza)
It has been proven in [GMS] that extreme Khovanov homology is a reduced homology of the simplicial complex obtained from bipartite circle graph constructed from a link diagram (so called Lando graph of a link). In this talk we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, this homotopy type is a link invariant. We prove the conjecture in many special cases and find it convincing to generalize conjecture for any circle graph (intersection graph of cords in a circle). As suggested by M.Adamaszek, we prove that for a permutation graph the conjecture hold. We demonstrate how, for any finite wedge of spheres, to find a permutation graph G with IG of its homotopy type. We modify our construction to show, in particular, that for any n there is a Lando graph G with its independence simplicial complex, IG, homotopy equivalent to Sn∨S2n-1. Another family of graphs, again suggested by Adamaszek, have non-nested cords on one side of the circle. We prove that for this family (and its small generalizations) the conjecture holds. We give several other examples supporting wedge of spheres conjectures, but full conjectures are still open. We would like to thank Michal Adamaszek, Sergei Chmutov, and Victor Reiner for helpful advise.
[GMS] J.González-Meneses, P.M.G.Manchón, M.Silvero, A geometric description of the extreme Khovanov cohomology, e-print: arXiv:1511.05845 [math.GT]
Date received: April 14, 2016
Exceptional Cosmetic Surgeries on S3.
by
Huygens Ravelomanana
University of Georgia
Two distinct Dehn surgeries on the same knot are called cosmetic if they produces homeomorphic 3-manifolds. The knot cosmetic surgery problem asks if cosmetic surgeries do exist. One of my result is that the slope of an exceptional truly cosmetic surgery (if it exists) on a hyperbolic knot in S3 must be ±1 and the surgery must be irreducible and toroidal but not Seifert fibred. This restricts the type of knots which can admit exceptional truly cosmetic surgeries. In particular there are no exceptional truly cosmetic surgeries on alternating hyperbolic knots in S3. There are no exceptional truly cosmetic surgeries on arborescent knots in S3.
Date received: February 17, 2016
Counting Cycles in Directed Graphs
by
Thomas Riggs
George Washington University
In any directed graph, the cycle packing number is the maximum amount of edge disjoint directed cycles
within the graph. In my research, I developed an extension to the Macauly2 software package,
which allows a user to input a directed graph and receive the cycle packing number. This talk is
based on my senior thesis, which was done under the guidance of Professor Hao Wu of The George Washington
University.
Date received: April 12, 2016
On a possible relation between HOMFLY-PT homology and geometric Langlands duality
by
Lev Rozansky
Department of Mathematics, University of North Carolina
Coauthors: Alexei Oblomkov
We will explain a possible relation between (1) the HOMFLY-PT homology, (2) the sheaves on a Hilbert scheme of C^2 and (3) a certain construction within a Kapustin-Witten TQFT, which is a part of their explanation of the geometric Langlands duality.
Date received: March 27, 2016
Cocycle knot invariants with topological quandles
by
Masahico Saito
University of South Florida
Coauthors: W. Edwin Clark
Quandle cocycle invariants were defined for knots using finite quandles. We generalize the 2-cocycle invariant for classical knots to topological quandles. The generalized invariants are computed explicitly for some knots, using SO(3) with quandle structures called generalized Alexander quandles.
Date received: April 15, 2016
Torsion in Khovanov link homology via chromatic graph cohomology
by
Dan Scofield
NCSU
The categorification of the chromatic polynomial by Helme-Guizon and Rong is isomorphic to Khovanov link homology over a range of homological gradings. Motivated by Hochschild homology, we compute torsion in chromatic homology for certain classes of graphs. As a consequence, we offer insight into Z2 torsion of certain classes of knots and links.
Date received: April 23, 2016
On coherent regions of a oriented knot diagram
by
Reiko Shinjo
Kokushikan university
Coauthors: Kokoro Tanaka
Let D be an oriented knot diagram on the two sphere. A face of D is called a coherent (resp. incoherent) region if the orientation of its boundary is coherent (resp. incoherent). We gave some relations between the number of the incoherent regions and the canonical genus of a knot. In this talk, we characterize the knots with a diagram which has less than four coherent regions. This is joint work with Kokoro Tanaka.
Date received: April 17, 2016
The AJ conjecture for cable knots
by
Anh T Tran
University of Texas at Dallas
The AJ conjecture relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. In this talk, we will discuss the conjecture for cable knots. Using skein theory, we will show that the AJ
conjecture holds true for most (r,2)-cables of some classes of two-bridge knots and pretzel knots.
Date received: March 18, 2016
Brunnian braids and Lie algebras
by
Vladimir Vershinin
Université de Montpellier, France
Coauthors: J. Y. Li and J. Wu
Brunnian braids have interesting relations with homotopy groups of spheres. In this work, we study the graded Lie algebra of the descending central series related to Brunnian subgroup of the pure braid group. We prove that this is a free Lie algebra and we find its set of free generators. We also calculate the rank of this algebra in each grading.
Date received: March 31, 2016
Geometry of secants
by
Oleg Viro
Stony Brook Universtiy
A secant of a link is a line which intersects the link. Secants of a link form a space stratified by the number of intersection points as well as by other geometric properties of secants. If the link is real algebraic curve, then the stratification is refined by interaction between the complexifications of the link and secants. Strata are equipped with functions that are naturally defined and related to link invariants.
In the talk we will study the strata, their modifications under isotopies and formulas relating characteristics of the strata to each other and to known link invariants.
Date received: April 24, 2016
On the sl2 Weight System and Intersection Graphs
by
Huan Vo
University of Toronto
Coauthors: Dror Bar-Natan
Given a chord diagram D, the value of the sl2 weight system on the primitive part of D is a polynomial in c, the Casimir element of sl2. It turns out that the coefficient of the highest power of c can be computed in terms of the intersection graph of D. This formula was first conjectured in a paper by Lando et al. In this talk, I will describe the formula, and if time permits, sketch a proof of it, which is a simple consequence of the Melvin- Morton-Rozansky conjecture.
Date received: March 17, 2016
Milnor invariants of covering links
by
Kodai Wada
Waseda University
Coauthors: Natsuka Kobayashi, Akira Yasuhara
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartely and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant for a link, and that this invariant can distinguish some links for which the ordinary Milnor invariants coincide. Moreover, for a Brunnian link L, the first non-vanishing Milnor invariants of L is modulo-2 congruent to a sum of Milnor invariants of covering links.
Date received: March 9, 2016
Higher Order Stability in the Colored Jones Polynomial
by
Katherine Walsh
University of Arizona
We will discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we show an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots and determine which knots have the same higher order stability. Conjectures about what the third stable sequence can tell us about the knot's geometry will also be presented.
Date received: March 22, 2016
Equivalence of "graphic" and älgebraic" definitions of set-theoretic Yang-Baxter homology
by
Xiao Wang
The George Washington University
Coauthors: Jozef Przytycki
In 2004, Carter, Elhamdadi and Saito defined a homology theory for the set-theoretic Yang-Baxter operator. In 2012, Przytycki defined another homology theory for Yang-Baxter operator which has a nice graphic visualization. We show that they are equivalent in the sense that they give the same homology group.
Date received: April 26, 2016
Annular homology and Hochschild homology
by
Ben Webster
University of Virginia
I'll discuss how to extract annular Khovanov(-Rozansky) homology and generalizations of these invariants for other groups from categorifications of tensor products, using Hochschild homology. This generalizes recent work of Queffelec-Rose relating the horizontal and vertical traces of categorified quantum groups.
Date received: April 17, 2016
Geometric realizations of distributive structure homology and shadow homotopy invariants.
by
Seung Yeop Yang
The George Washington University
Coauthors: Jozef H. Przytycki
In 1993, Fenn, Rourke, and Sanderson introduced rack spaces and constructed rack homotopy invariants, and a modication to quandle spaces and quandle homotopy invariants of classical links was introduced by Nosaka in 2011.
In analogy to rack and quandle spaces, we define the Cayley-type graph and CW complex of a distributive structure and study their properties. Moreover, for a quandle we introduce the shadow homotopy invariant of classical links.
Date received: April 19, 2016
Copyright © 2016 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.