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Organizers |
Colored Khovanov-Rozansky homology of infinite braids
by
Michael Abel
Duke University
Coauthors: Michael Willis (University of California, Los Angeles)
We show that the limiting unicolored sl(N) Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp.
Date received: November 18, 2017
Ribbon Obstructions and Singular Branched Covers of Four-Manifolds
by
Patricia Cahn
Smith College
Coauthors: Alexandra Kjuchukova (UW-Madison)
Consider a four-manifold Y which is presented as a p-fold dihedral branched cover of S4, with one singularity on the branching set, modelled on the cone on a knot K. Kjuchukova showed that the signature of Y is an invariant of K. We show that this signature is a ribbon obstruction, and give an algorithm for computing this signature from a p-colored knot diagram of K. We use trisections to identify the diffeomorphism type of the cover for given families of singularities. In particular, we construct infinitely many singular dihedral covers of S4 by CP2. We conclude by giving a classification of singular dihedral branched covering maps from CP2 to S4, and explain the implications of this classification for finding potential counterexamples to the Slice-Ribbon Conjecture.
Date received: November 24, 2017
A Prismatic Classifying Space
by
J. Scott Carter
University of South Alabama
Coauthors: Victoria Lebed, Seung Yeop Yang
A qualgebra G is a set having two binary operations that satisfy compatibility conditions which are modeled upon a group under conjugation and multiplication. We develop a homology theory for qualgebras and describe a classifying space for it. This space is constructed from G-colored prisms (products of simplices) and simultaneously generalizes (and includes) simplicial classifying spaces for groups and cubical classifying spaces for quandles. Degenerate cells of several types are added to the regular prismatic cells; by duality, these correspond to "non-rigid" Reidemeister moves and their higher dimensional analogues. Coupled with G-coloring techniques, our homology theory yields invariants of knotted trivalent graphs in R3 and knotted foams in R4. We re-interpret these invariants as homotopy classes of maps from S2 or S3 to the classifying space of G.
Date received: November 16, 2017
Chord index and its applications in virtual knot theory
by
Zhiyun Cheng
Beijing Normal University
In this talk, I will give a brief introduction to the chord index of virtual knots. Several applications of the chord index will be given. Finally, I would like to discuss how to use a biquandle to generalize the notion of chord index.
Date received: September 14, 2017
Some Corollaries of Manturov's projection Theorem
by
Vladimir Chernov
Dartmouth College
In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus gvc(K) and the virtual bridge number vb(K) invariants of virtual knots. One can see from the definitions that for an ordinary knot K the values of these invariants are less or equal than the classical canonical genus gc(K) and the bridge number b(K) respectively. We use Manturov's projection from the category of virtual knot diagrams to the category of ordinary knot diagrams, to show that for every ordinary knot type K we have gvc(K)=gc(K) and vb(K)=b(K).
Date received: November 25, 2017
An introduction to Spiders and Projectors
by
Amanda Curtis
UCSB / The Maderia School
Within spiders and planar algebras exist a special set of diagrams called projections. These are diagrams p for which p2=p. In his dissertation, Kim provides a recursive formula for the SL(3) planar algebra. We provide a new formula for them which is not recursive, and consider some other important diagrams for this algebra as well.
Date received: December 3, 2017
Complexity of Virtual Multistrings
by
David Freund
Dartmouth College
A virtual n-string α is a collection of n closed curves on an oriented surface M. Associated to α, there are two natural measures of complexity: the genus of M and the number of intersection points. By considering virtual n-strings up to equivalence by virtual homotopy, i.e., homotopies of the component curves and stabilizations/destabilizations of the surface, a natural question is whether these quantities can be minimized simultaneously. We show that this is possible for non-parallel virtual n-strings and that, moreover, such a representative can be obtained by monotonically decreasing genus and the number of intersections from any initial representative.
Date received: October 19, 2017
Order Relations on Computable Magmas
by
Trang Ha
GWU
Coauthors: Valentina Harizanov
A magma is computable if there is an algorithm to decide the membership of its elements and the magmas binary operation is computable. We discuss order relations on computable magmas. We will show conditions for a magma to be orderable, Turing degrees, and description of the space of orders. We also consider orderings on some examples of non associative magmas such as quandles and racks.
Date received: December 5, 2017
Unicity for Representations of the Kauffman Bracket Skein Algebra
by
Joanna Kania-Bartoszynska
National Science Foundation
Coauthors: Charles Frohman and Thang Le
Let F be a connected, closed, oriented surface, with finitely many (possibly none) points removed, and let ζ be a root of unity. The Kauffman bracket skein algebra of F, denoted Kζ(F), is formed by taking C-linear combinations of isotopy classes of links in a cylinder over F, and modding out by the Kauffman bracket skein relation, with the parameter ζ. Multiplication is given by placing one link over another and extending linearly. Bonahon and Wong associate to each irreducible representation of Kζ(F) a classical shadow, which is determined by its central character, and conjecture that there is a generic family of classical shadows for which there is a unique irreducible representation of the skein algebra realizing that classical shadow. In joint work with Charles Frohman and Thang Le we resolve this conjecture. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine k-algebra over an algebraically closed field k, that is finitely generated as a module over its center, are generically classified by their central characters.
Date received: December 5, 2017
Knotoids, Virtual Knots and Applications
by
Louis H Kauffman
University of Illinois at Chicago
Coauthors: Neslihan Gugumcu, Dimos Goundaroulis, Sofia Lambropoulou
With Neslihan Gugumcu we study invariants of knotoids (open ended knot diagrams
with the endpoints in possibly different regions). These invariants include the bracket
polynomial,the Turaev loop bracket polynomial (for planar knotoids), the arrow
polynomial and a loop arrow polynomial. We discuss properties of these invariants,
and with Dimos Goundaroulis and Sofia Lambropoulou we apply these techniques
to the structure of open ended polymer strings and to protein folding.
Date received: August 30, 2017
Signed Cyclic Graphs, Checkerboards and Virtual Links
by
Louis H Kauffman
University of Illinois at Chicago
Abstract: This talk is joint work with Qingying Deng.
A signed cyclic graph is a graph signs on its edges and cyclic orders for each node.
We show that signed cyclic graphs are in 1-1 correspondence with checkerboard colorable virtual link diagrams.
We give a graphical model for the bracket polynomial of the virtual link associated with a signed cyclic graph and compare with the signed Tutte polynomial.
We show how virtualization of crossings translates in the signed graph category and discuss problems about checkerboard colorings and free knots.
Other problems will be discussed depending on the time.
Date received: October 30, 2017
Categorification of integers with two inverted
by
Mikhail Khovanov
Columbia University
Coauthors: Yin Tian
We will describe a monoidal triangulated category, closed with respect to taking idempotents, and with the Grothendieck ring isomorphic to the ring of integers localized at two.
Date received: December 3, 2017
1-stable equivalence of knotted surfaces in 4-manifolds
by
Hee Jung Kim
Seoul National University
Coauthors: Dave Auckly (Kansas State University), Paul Melvin (Bryn Mawr College), Daniel Ruberman (Brandeis University), Hannah Schwartz (Bryn Mawr College)
In 4-dimensional topology, the existence of exotic smooth structures on closed simply-connected 4-manifolds derives from work of Donaldson and Freedman, and it is known by Wall that such manifolds become diffeomorphic after stabilizing, i.e. connected summing with a S^2-bundle over S^2, sufficiently many times. The question of the minimal number of stabilizations to be diffeomorphic arises and it has turned out that a single stabilization suffices for all known examples.
In this talk, we will discuss an analogue of this principle related to embedded surfaces in 4-manifolds. In particular, we will show that the 1-stable equivalence principle holds for surfaces with simply-connected complements; any two homologous surfaces of the same genus embedded in a smooth 4-manifold with simply-connected complements are smoothly isotopic after 'single' stabilization with a S^2-bundle over S^2.
Date received: November 20, 2017
Applications of TQFT to the structure of the Yamada and flow polynomials
by
Slava Krushkal
University of Virginia
I will discuss several applications of TQFT to the structure of the chromatic and flow polynomials of
(planar and non-planar) graphs, and of the Yamada polynomial of ribbon graphs in 3-space.
This talk is based on joint work with Ian Agol.
Date received: December 4, 2017
Circuits and Hurwitz action in finite root systems
by
Joel Brewster Lewis
GWU
Coauthors: Victor Reiner (UMN)
There is a natural braid group action on tuples of elements in a group, called the Hurwitz action. In a finite real reflection group, all factorizations of a Coxeter element into a minimal number of reflections lie in the same orbit of this action. We extend this result to factorizations of arbitrary length, showing that two reflection factorizations of a Coxeter element lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.
Date received: November 28, 2017
The search for simplicity in higher dimensional knot theory.
by
Samuel J. Lomonaco
University of Maryland Baltimore County (UMBC)
We discuss an ongoing research program to find an efficient algorithm (similar to Wirtinger's for 1-knot groups) to compute the algebraic 2-type of the exterior X of a 2-knot (S^4,kS^2), i.e., the triple consisting of Pi_1(X), Pi_2(X) as a ZPi_1(X)-module, and the k-invariant k(X) lying in H^3(Pi_1(X),Pi_2(X)).
Date received: November 26, 2017
The tunnel number of handlebody-knots and G-families of biquandles
by
Tomo Murao
Tsukuba University
A handlebody-knot is a handlebody embedded in the 3-sphere S3. The tunnel number of a handlebody-knot is the minimal number of 1-handles that must be attached to the handlebody-knot such that the complement becomes a handlebody. In this talk, we give an evaluation formula of the tunnel number of handlebody-knots by using G-families of biquandles and some examples.
Date received: September 27, 2017
Psyquandles, Singular Knots and Pseudoknots
by
Sam Nelson
Claremont McKenna College
Coauthors: Natsumi Oyamaguchi and Radmila Sazdanovic
Psyquandles are an algebraic structure with axioms motivated by the Reidemeister moves for singular knots and pseudoknots. In this talk we will introduce psyquandles, give examples of finite psyquandles and the psyquandle counting invariant, and introduce an Alexander polynomial-style invariant of singular knots and pseudoknots we call the Jablan polynomial.
Date received: October 17, 2017
Quantization of the annular Khovanov homology
by
Krzysztof Putyra
University of Zurich
Coauthors: Anna Beliakova, Stephan M. Wehrli, Matthew Hogancamp
I will discuss a deformation of the annular Khovanov homology that carries an action of the quantum sl(2). The first step is to construct an isomorphism between undeformed annular Khovanov homology and Hochschild hyperhomology of the Chen-Khovanov invariant of tangles, conjectured by Auroux, Grigsby, and Wehrli. Rephrasing this in the language of categorical traces allows us to recover the sl(2) action due to Grigsby, Licata, and Wehrli, and then deform it. The new invariant has many interesting properties. For instance, it assigns a nontrivial polynomials to knotted surfaces in S1 x D3 and the quantized colored Khovanov complex is homotopy equivalent to the CooperKrushal complex (so that both are finite dimensional).
This is a joint project with Anna Beliakova, Stephan Wehrli, and Matthew Hogancamp.
Date received: December 4, 2017
On factorization and chromatic graph homology
by
Radmila Sazdanovic
NC State
Coauthors: Vladimir Baranovsky
Factorization homology, introduced by Ayala, Francis, and Tanaka, generalizes Hochschild homology. Helme-Guizon and Rongs chromatic graph homology of a circle approximates Hochschild homology. We show that chromatic homology can be obtained in a similar way as factorization homology. The main difference between the two constructions stems from using derived versus underived products. Therefore the chromatic homology of any graph can be thought of as an approximation of factorization homology.
Date received: December 5, 2017
Fiberwise isotopies of toral surfaces
by
Jonathan Schneider
Saint Xavier University
Fiberwise-embedded tori in 4-space are related via fiberwise isotopies. We visualize these isotopies as sequences of Roseman moves under projection.
Date received: November 4, 2017
Semiadequate links and second to extreme Khovanov homology.
by
Marithania Silvero
Institute of Mathematics of the Polish Academy of Sciences
Coauthors: Józef H. Przytycki
Given a semiadequate diagram D representing a link L, we present an algorithm for constructing a presimplicial set such that its geometric realization is homotopy equivalent to the almost-extreme Khovanov complex of L. Moreover, we determine the homotopy type of the presimplicial set obtained when the link is strongly A-adequate. We show explicitly the particular cases of trefoil and figure-8 knots.
Date received: November 19, 2017
On meta-monoids and the Alexander polynomial
by
Huan Vo
University of Toronto
We define an algebraic structure called meta-monoids which are suited for the study of knot theory. We introduce a particular meta-monoid called Γ-calculus which yields an Alexander invariant of tangles. Using Γ-calculus, we rederive certain basic properties of the Alexander polynomial, most notably the Fox-Milnor condition for ribbon knots. This talk is based on the paper arXiv:1710.08993
Date received: November 6, 2017
Torsion in Khovanov Homology and links with smoothing number one
by
Xiao Wang
The George Washington University
Coauthors: Sujoy Mukherjee, Józef H. Przytycki, Marithania Silvero and Seung Yeop Yang
In the Khovanov homology of links, presence of Z2-torsion is a very common phenomenon. Finite number of examples of knots with Zn-torsion for n > 2 were also known, none for n > 8. In this talk, we demonstrate that there are infinite families of links with smoothing number one, whose Khovanov homology contains Zn-torsion for 2 < n < 9 (Tk(n, n+2)) and Z2s-torsion for s < 24 (Tk(4, 4s+2)). The idea of the proof also works for other families, for instance Tk(n, n+1). We also provide an infinite family of links with Z5-torsion in reduced Khovanov homology and Z3-torsion in odd Khovanov homology. Finally, I will mention the possibility of computing the Khovanov homology of links with smoothing number one through Hochschild homology. This is a joint work with Mathathoners.
Date received: December 4, 2017
A rank inequality for the annular Khovanov homology of 2-periodic links
by
Melissa Zhang
Boston College
Date received: November 29, 2017
Copyright © 2017 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.