Topology Atlas | Conferences


Knots in Washington XLI
December 4-6, 2015
George Washington University
Washington, DC, USA

Organizers
Valentina Harizanov (GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (NCSU), Alexander Shumakovitch (GWU), Hao Wu (GWU)

Conference Homepage


Abstracts

The logic of partial information and Bohr's complementarity principle in Quantum Mechanics
by
Areski Nait Abdallah
Univ. of Western Ontario, Canada and INRIA Rocquencourt, France

Starting from Grangier and Aspect's 1986 landmark experiments on single-photon interferences, we review Bohr's complementarity principle in Quantum Mechanics  and the related wave-particle duality paradox.

We show how the wave-particle duality paradox is resolved using lambda-calculus and the logic of partial information. The logic of partial information was originally designed specifically for the needs of practicing computer scientists reasoning with partial information, and aimed at providing a formalization of Popper and Lakatos' logic of scientific discovery.

Date received: November 17, 2015


Stable homology of torus links via categorified Young antisymmetrizers
by
Michael Abel
Duke University
Coauthors: Matt Hogancamp

We construct complexes of Soergel bimodules which categorify the Young antisymmetrizers in Hecke algebras. A beautiful recent conjecture of Gorsky and Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for Young antisymmetrizers and their twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of (n,nm+k) torus links.

Date received: November 16, 2015


Dualities and Trialities from Ribbon Group Stabilizers
by
Lowell Abrams
George Washington University
Coauthors: Jo Ellis-Monaghan (Saint Michael's College)

The ribbon group acts on cellulary embedded graphs by applying Wilson operations, generated by dualizing and twisting, to individual edges. To account for actions on edge-labeled graphs, we reformulate this action in terms of a semidirect product with the symmetric group on the ribbons. In this context, we show how studying the stabilizers of single-vertex orientable embeddings provides an approach to produce, for example, self-dual and self-trial cellular embeddings that are not necessarily regular. We close with discussion of some results of a computer search for examples.

Date received: November 18, 2015


Obstructing Sliceness in Odd Pretzel Knots
by
Kathryn Bryant
Bryn Mawr College
Coauthors: Paul Melvin

We apply techniques from Greene and Jabuka's 2011 paper on the Slice-Ribbon Conjecture for odd, 3-stranded pretzel knots to say the following: (1) All odd pretzel knots having a difference of 2 or more in the number of positive twist parameters and the number of negative twist parameters have infinite order in the smooth knot concordance group, and (2) 0-Pair, odd, 5-stranded pretzel knots are not slice.

The proofs involve the classical slice obstruction coming from the knot signature, as well as more modern slice obstructions coming from Donaldson's Diagonalization Theorem and d-invariants from Heegaard-Floer homology.

Date received: October 27, 2015


Index type invariants of virtual knots
by
Zhiyun Cheng
Beijing Normal University, George Washington University

In this talk I will give a short survey on virtual knot invariants defined by assigning an index to each real crossing. Some applications and some recent progress will be discussed.

Date received: November 17, 2015


Universal Orderability of Legendrian Isotopy Classes
by
Vladimir Chernov
Dartmouth College
Coauthors: Stefan Nemirovski

It is shown that non-negative Legendrian isotopy defines a partial order on the universal cover of the Legendrian isotopy class of the fibre of the spherical cotangent bundle of any manifold. This result is applied to Lorentz geometry in the spirit of the authors’ earlier work on the Legendrian Low conjecture.

Date received: September 29, 2015


Formal Contact Categories
by
Ben Cooper
University of Iowa

To each oriented surface S we associate a differential graded category. The homotopy category is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are related to the algebraic contact categories of Y. Tian and the bordered sutured categories of R. Zarev.

Date received: November 18, 2015


Bounding the topological slice genus using the Seifert pairing.
by
Peter Feller
Boston College

We present upper bounds for the topological slice genus of knots and links given by a simple criterion on their Seifert form. The main ingredient in the proof is Freedman's disk theorem.

The criterion yields surprising consequences for simple families of knots that contrast smooth results such as the Thom conjecture.

Parts of the talk are based on joint work with McCoy and Baader, Lewark, and Liechti.

Date received: October 19, 2015


Quantum in variants of three manifolds equipped with an SL(2, C) representation of its fundamental group.
by
Charles Frohman
University of Iowa

We begin by reviewing the structure of the Kauffman Bracket Skein Algebra at a root of unity,

And it's relation to the ring of SL(2,C) characters of the fundamental group of the surface.

Next we show how this gives rise to invariants of three-manifolds that fiber over the circle.

Finally, we outline how this can give rise to a field theory.

Date received: September 28, 2015


Signature Jumps and Alexander Polynomials for Links
by
Patrick M. Gilmer
Louisiana State University
Coauthors: Charles Livingston

Given a spanning surface for an oriented link, one may associate a family of Hermitian forms which are parametrized by the points on the unit circle in the complex plane. The signatures of these forms do not depend on the choice of a spanning surface, and thus define an integer valued function on the unit circle. It is a step function. The spanning surface can also be used to define a sequence of polynomials called the Alexander polynomial and higher Alexander polynomials of the link.

A link with one component is a knot. For knots, it is well-known that the signature function can only jump at roots of the Alexander

polynomial and the jump at a root is less or equal to the multiplicity of the root.

What can one say if the link has more than one component? Then it may happen that the Alexander polynomial is identically zero. It turns out that a similar result holds about the location and size of the jumps if one replaces the Alexander polynomial with the first non-vanishing higher Alexander polynomial.

(This talk is also a colloquium, 1:00pm Friday, Dec 4, 2015.)

Date received: September 29, 2015


Two Generalizations of the Multivariable Alexander Polynomial
by
Iva Halacheva
University of Toronto

An extension of the Multivariable Alexander Polynomial (MVA) for links to virtual tangles, taking values in a tensor product of exterior algebras, was defined by J. Archibald in her thesis (arXiv:0710.4885v1). The computations involve and exponential-time algorithm but are relatively straightforward and provide an easy verification of almost all the relations satisfied by the MVA and its weight system. Another generalization of the MVA is given by D. Bar-Natan and comes from a reduction of an invariant of knotted copies of S2 and S1 in four-dimensional space. It has the advantage that it is matrix-valued and computable in polynomial time, but is only defined on tangles with no closed components, i.e. pure tangles. We will show that after some repackaging, the two invariants coincide on the level of pure tangles, and we will discuss a partial extension of Bar-Natan’s invariant, arising from this connection, which allows closed components.

Date received: October 22, 2015


Asymmetric knots with two cyclic surgeries
by
Neil R Hoffman
University of Melbourne
Coauthors: Nathan M. Dunfield, Joan E Licata

The cyclic fillings of a hyperbolic manifold are of considerable interest. John Berge constructed a list of knots in S3 that admit a non-trivial cyclic filling and the Berge conjecture states that this list is complete. A consequence of the Berge Conjecture is that all such knot complements admit an order two symmetry. While the natural generalization of the Berge Conjecture still provides a list of hyperbolic manifolds with two cyclic fillings, we show such a list is incomplete. Finally, this provides examples of L-spaces which are not the double branched covers of knots in S3.

Date received: October 20, 2015


Braid rank and detecting quasipositivity of braids.
by
Mark Hughes
BYU

Abstract: We describe a new technique to recast geometric problems involving the 4-ball and ribbon genera of a link in terms of algebraic properties of a braid representative. These techniques make use of braided cobordisms, and require the study of certain shortest word problems in the braid group described by Rudolph. This leads to a new algebraic invariant of the knot. We will present an upper bound to the solution of this shortest word problem, and discuss the related problem of detecting quasipositive braids.

Date received: November 28, 2015


The Kauffman Polynomial of Periodic Links
by
Kyle Istvan
Louisiana State University
Coauthors: Khaled Qazaqzeh, Ayman Aboufattoum

A periodic link has a diagram that is invariant under a finite-order rotation in the plane. I will define a necessary condition for a link to be p-periodic, for odd prime p. It takes the form of a congruence between a specialization of the 2-variable Kauffman polynomial of a link and that of the link's mirror image. The result is derived using a state sum formula for the 2-variable polynomial, and can be used to verify (for example) Traczyk's result that the knot 10101 is not 7-periodic.

Date received: October 29, 2015


Every empire has a defendable border region
by
Paul C. Kainen
Georgetown University

The title is a colloquial description of the following Lemma: For every connected plane graph G and every cycle z of G, either z is itself the boundary of a region or else there is a region r of the graph such that r is contained inside z and the boundary of r intersects z in a set homeomorphic to the closed unit interval I. This implies that the Mac Lane cycle basis of a plane graph is "robust" - i.e., if z is a cycle, the basis cycles, of which z is the sum, can be ordered such that each successive cycle after the first intersects the mod-2 sum of the previous terms in a homeomorph of I. Hence, all of the partial sums remain cycles. Implications for the algebra of commutative diagrams will be briefly discussed.

Date received: November 13, 2015


Structure of the Kauffman bracket skein algebra of a surface
by
Joanna Kania-Bartoszynska
National Science Foundation

Kauffman bracket skein algebra of a surface is formed by taking linear combinations of isotopy classes of framed links in a cylinder of the surface and dividing by the relation which defines the Kauffman bracket. Multiplication comes from stacking one link over the other. Kauffman bracket skein algebras are related to the SL(2, C) characters of the fundamental group of the surface. They are used in skein theoretic constructions of topological quantum field theories. It turns out that those algebras are integral domains, and that they are Frobenius when localized over non-zero characters.

Date received: November 28, 2015


Rotational Virtual Links and Quantum Link Invariants
by
Louis H Kauffman
University of Illinois at Chicago

This talk is about rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We set up the background virtual knot theory, define rotational virtual knot theory, study an extension of the bracket polynomial and the Manturov parity bracket for rotationals. Then the general frameworks for oriented and unoriented quantum invariants are introduced and formulated for rotational virtual links. The talk ends with quantum link invariants in the Hopf algebra framework where one can see the naturality of using regular homotopy combined with virtual crossings (permutation operators), as they occur significantly in the category associated with a Hopf algebra. We show how this approach via categories and quantum algebras illuminates the structure of invariants that we have already described via state summations. In particular, we show that a certain link L has trivial functorial image. This means that this link is not detected by any quantum invariant formulated as outlined here. At this writing, we conjecture that the link L is a non-trivial rotational link. These calculations with the diagrammatic images in quantum algebra show how this category forms a higher level language for understanding rotational virtual knots and links. If the conjecture is true, then there are inherent limitations to studying rotational virtual knots by quantum algebra alone.

Date received: October 19, 2015


Homotopy Theory of Link Homology via the Dold-Kan Theorem
by
Louis H Kauffman
UIC
Coauthors: Chris Gomes

The Dold-Kan constructon in simplicial homotopy theory can be applied to convert link homology theories into homotopy theories. We construct a mapping F: L → S taking link diagrams L to a category of simplicial spaces S such that up to looping or delooping, link diagrams related by Reidemeister moves will give rise to homotopy equivalent simplicial objects, and the homotopy groups of these objects will be equal to the link homology groups of the original link homology theory. The construction is independent of the particular link homology theory, applying equally well to Knovanov Homology and to Knot Floer Homology and other theories of these types. The construction is of particular interest for Khovanov Homology where there is a natural pre-simplicial structure already present in the definition of the Khovanov category. This allows us to define an embedding of the cube category into a simplicial category so that the map to Frobenius algebras determined by a knot or link produces a simplicial module. The homology of this simplicial module is Khovanov homology. The homotopy type of this simplicial module is the homotopy type to which we refer above.

(Talk to be given on Monday, December 7, 2015.)

Date received: October 20, 2015


Linearization and categorification
by
Mikhail Khovanov
Columbia University

This is a light talk explaining linearization through examples and categorification that sometimes follows it.

Date received: November 21, 2015


Geometric complexity of embeddings
by
Slava Krushkal
University of Virginia

There are various notions of geometric complexity of embeddings: thickness, distortion, the minimal number of subdivisions needed for a PL embedding. I will discuss several problems concerning complexity of knots and links in 3-space, and complexity of embeddings in higher dimensions.

Date received: November 24, 2015


On the arc index of cable links and Whitehead doubles
by
Hwa Jeong Lee
KAIST (Korea Advanced Institute of Science and Technology)
Coauthors: Hideo Takioka

In this talk, we construct an algorithm to produce "canonical grid diagrams" of cable links and Whitehead doubles. We show that the algorithm correctly determines the arc index of 2-cable links and Whitehead doubles of all prime knots with up to 8 crossings.

Date received: November 16, 2015


Higher Dimensional Knot Theory: How I stumbled upon the results
by
Samue Lomonaco
University of Marland Baltimore County (UMBC)

In this talk, we discuss how the speaker stumbled upon different research results in higher dimensional knot theory. In particular: 1) How the Fox Free Derivative magically popped up in a long calculation of the second homotopy group of a spun knot. 2) How a long calculation resulted in a method for reducing four dimensional knot theory to three dimensional knot theory. 3) How the above appearance of the Fox Free Derivatives surprisingly morphed into the discovery of a Generalized Eilenberg MacLane (GEM) complex.

Date received: December 1, 2015


Signatures and Turaev Genera of Knots
by
Adam Lowrance
Vassar College
Coauthors: Oliver Dasbach

We will discuss formulas for the signature of knots with small Turaev genus.

Date received: October 10, 2015


Extracting integer invariants from a power series expansion of the Jones polynomial
by
Vajira Manathunga
University of Tennessee, Knoxville

It is known that appropriate change of variable of Jones polynomial followed by Taylor series expansion gives an infinite power series with coefficients that are Vassiliev invariants. However these Vassiliev invariants are rational valued. We can convert them to integer valued Vassiliev invariants by multiplying it with appropriate constant λk. In this talk we give a formula for minimal λk when k is even and some interesting congruence relationships between these invariants.

Date received: November 3, 2015


Biquandle Brackets
by
Sam Nelson
Claremont McKenna College
Coauthors: Michael E. Orrison (HMC) and Veronica Rivera (HMC)

Given a finite biquandle X and a commutative ring with identity R, we define an algebraic structure known as a biquandle bracket. Biquandle brackets can be used to define a family of knot and link invariants known as quantum enhancements which include biquandle cocycle invariants and skein polynomials such as the Alexander, Jones and HOMFLYpt polynomials as special cases. As an application we will see a new skein invariant which is not determined by the knot group, the knot quandle or the HOMFLYpt polynomial.

Date received: September 28, 2015


Recent results concerning bridge spectrum
by
Nicholas Owad
University of Nebraska-Lincoln

The bridge spectrum of a knot is a generalization of the classic invariant defined by Schubert, the bridge number of a knot. We will introduce the relevant background and some known results. Then we will state our main result, and end with open questions.

Date received: October 27, 2015


3D printing for knot theory and topology
by
Nicholas Owad
UNL
Coauthors: Kyle Istvan

In recent years, the cost of 3D printing has dropped significantly, to the point where it is more economical for departments to fabricate their own models than order them online. We will discuss the basics of 3D printing and the various technologies that are available. We'll also mention some of the mathematical tools available to help generate printable models, and include examples of links, Seifert surfaces, and other familiar examples from low-dimensional topology.

Date received: November 20, 2015


On categorical traces and homology for links in a solid torus
by
Krzysztof K Putyra
ETHZ Institute for Theoretical Studies
Coauthors: Anna Beliakova

A trace of a category is the set of its endomorphisms considered up to conjugation. For example, the trace of the category of tangles is the set of links in a solid torus: every such a link is a closure of a tangle and closures of two tangles coincide if and only if the tangles agree up to conjugation. The categorical trace is functorial: a functor between categories induces a map between their traces. In particular, we can obtain invariants of links in a solid torus from functors on the category of tangles. In my talk I will discuss the trace of the tangle invariant due to Chen and Khovanov, which recovers the annular sl2 homology defined by Asaeda, Przytycki, and Sikora. This construction almost immediately equips the APS homology with an action of sl2. With a small modification of the trace construction one can quantize the homology to obtain a sequence of representations of quantum sl2.

Date received: November 3, 2015


Khovanov-Rozansky homology and current algebras
by
David E. V. Rose
University of Southern California
Coauthors: Hoel Queffelec

We'll use the sl_n foams developed by the speaker (joint with Queffelec) to show that the "Khovanov-Rozansky complex" of a link in S^3 presented as the closure of a balanced tangle admits a special representative in which the differentials are given by the action of a current algebra. This endows the complex with a filtration, and the homology of the associated graded complex is an invariant of the tangle closure, viewed as a link in the thickened annulus. Time permitting, we'll show how this structure naturally equips this "annular Khovanov-Rozansky homology" with an action of sl_n, extending a result of Grigsby-Licata-Wehrli.

Date received: November 29, 2015


Examples of virtual knots with vanishing n-writhes
by
Migiwa Sakurai
National Institute of Technology, Ibaraki College
Coauthors: Sumiko Horiuchi and Yoshiyuki Ohyama

Satoh and Taniguchi define a numerical invariant called an n-writhe. This invariant induces a lot of virtual knot invariants such as an index polynomial, an odd writhe polynomial and an affine index polynomial. In this talk, we show that there exists infinitely many virtual knots whose n-writhes are all zero.

Date received: October 6, 2015


Cultural Evolution of Material Knot Diversity
by
Lauren Scanlon
Durham University
Coauthors: Andrew Lobb, Jeremy Kendal, Jamie Tehrani

Knots are an important part of everyday life, tied for a variety of purposes including to secure our shoelaces. But how many different knots are actually used in everyday life? Are these knots optimised for their different uses? Why do we use the knots we do?

Using methods from Mathematics and Anthropology, I will discuss how we aim to answer these questions through building up a knot database and cultural experiments.

Date received: November 23, 2015


Parity Biquandle Invariants of Virtual Knots
by
Leo Selker
Pomona College
Coauthors: Sam Nelson (Claremont McKenna College), Aaron Kaestner (North Park University)

Given a biquandle X, we have a biquandle counting invariant and a biquandle cocycle invariant. In order to capture more structure of virtual knots, we use parity biquandles to construct the parity biquandle counting invariant. We also use parity biquandles to add additional structure to the cocycle invariant. We use a parity biquandle cocycle together with a compatible function to construct the parity-enhanced biquandle cocycle polynomial, which has the biquandle cocycle invariant as a special case. We consider some examples where these enhancements are nontrivial.

Date received: September 29, 2015


Complementary regions of knot diagrams and the canonical genus of knots
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). In this talk, we investigate the number of the coherent faces and incoherent faces of an oriented knot diagram, and give some relations between the number of the incoherent regions and the canonical genus of a knot. This is a joint work with Kokoro Tanaka (Tokyo Gakugei University)

Date received: November 25, 2015


Zero-divisors and central elements in skein algebras
by
Adam S. Sikora
SUNY Buffalo
Coauthors: Jozef H. Prytycki

For a surface F, the space of links in F×[0,1] modulo Kauffman bracket skein relations is called the skein algebra of F, denoted by S(F). It is a non-commutative deformation of the SL(2,C)-character variety of F, of significant importance to quantum topology. In particular, for F with boundary, it is (almost) the quantum Teichmuller space of F.

Except for a few simplest surfaces F, not much is known about the algebraic properties of S(F) for closed F. We are going to prove the following two fundamental properties of skein algebras: 1. S(F) has no zero divisors, 2. Away from roots of unity, the center of S(F) is composed of polynomials in knots parallel to boundary components of F. This is joint work with J. H. Przytycki.

Date received: November 29, 2015


A geometric realization of extreme Khovanov homology.
by
Marithania Silvero
Universidad de Sevilla (Spain)

In this talk we will give a geometric definition of the hypothetical extreme Khovanov homology of a link in terms of the cohomology of the independence complex of a special kind of graph constructed from the link.

Date received: October 21, 2015


Khovanov homology, independence complex and H-thick knots
by
Marithania Silvero
Universidad de Sevilla (Spain)

In this talk we will prove that the hypothetical extreme Khovanov homology of a link is the cohomology of the independence complex of its Lando graph. This result allows us to give examples of knots with an arbitrary number of non-trivial extreme Khovanov homology groups.

(Talk to be given on Thursday, December 3.)

Date received: October 21, 2015


A generalization of α-orientations to higher genus surfaces
by
Jason Suagee
George Washington University

Given a graph G=(V, E), and a function α:V→N, an α-orientation is an orientation of the edges such that the out-degree of each vertex v equals α(v). S. Felsner (TU Berlin) in 2004 proved that the set of α-orientation on an embedded planar graph (a planar map) carries the structure of a distributive lattice, with unique maximal and minimal elements. He uses this result, for example, to construct canonical spanning trees on rooted planar maps as well as several other canonical structures on planar maps.

We obtain a generalization of Felsner's result to higher genus orientable surfaces with possible application to bijective methods in map enumeration and construction. Additionally, by applying this result to pairs of Cayley maps (strongly symmetric embeddings of Cayley graphs) we obtain potential applications to the study of finite group extensions.

Date received: November 24, 2015


The cable Γ-polynomial of a knot
by
Hideo Takioka
Osaka City University Advanced Mathematical Institute

The Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. We consider the cable version of the Γ-polynomial. In this talk, we give sharper lower bounds of the braid indices of Kanenobu knots by applying the 2-cable Γ-polynomial. Moreover, we show that the 3-cable Γ-polynomial is invariant under mutation.

Date received: November 15, 2015


The canonical genus of Whitehead doubles of non-prime alternating knots
by
Kokoro Tanaka
Tokyo Gakugei University

The canonicai genus of a Whitehead double of a knot is less than or equal to its crossing number. Tripp observed that the equality holds for 2-braid knots and conjectured that the equality holds for all knots. However, Jang and Lee found counterexamples for this conjecture. In this talk, we discuss this conjecture for non-prime alternating knots.

Date received: November 29, 2015


Real algebraic knot theories
by
Oleg Viro
Stony Brook University

A classical knot may happen to be a real algebraic curve (i.e., be defined by a system of real polynomial equations). Any classical knot is isotopic to a real algebraic knot, but two real algebraic knots isotopic topologically may happen to be non-deformable to each other in the class of nonsingular real algebraic curves. Most phenomena studied in the classical knot theory have natural analogues in the theory of real algebraic knots. In the talk we will consider basic definitions of this theory, the first examples, invariants and classification results.

Date received: November 25, 2015


The Milnor invariants of clover links
by
Kodai Wada
Waseda University
Coauthors: Akira Yasuhara

J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most 2k+1 are well-defined if those of length at most k vanish, and that the Milnor numbers of length at least 2k+2 are not well-defined if those of length k+1 survive. For a clover link c with the Milnor numbers of length at most k vanishing, we show that the Milnor number μc(I) for a sequence I is well-defined up to the greatest common devisor of μc(J)'s, where J is a subsequence of I obtained by removing at least k+1 indices. Moreover, if I is a non-repeated sequence with length 2k+2, the possible range of μc(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links.

Date received: October 7, 2015


On the genus of a graph
by
Liangxia Wan
Beijing Jiaotong University

Determining the genus of a graph can be dated to the Heawood Conjecture in 1890. The Conjecture is implied by the Thread Problem by Hilbert and Cohn-Vossen which is the genus problem of a complete graph. A current graph was introduced by Gustin and a generalized current graph was done by Youngs in 1963. Ringel classified the genera of complete graphs and then this problem was solved in 1968.

Later, researchers mainly studied the genera of other special graphs with certain symmetry which are complete bipartite graphs, complete multipartite graphs n -cube etc. Until now the genera of complete tripartite graphs are not yet fully determined. In fact, Thomassen has proved that determining the genus of a graph( even a cubic graph) is NP-complete.

In this talk a formal set and its accompanying graph are introduced and a planarity rule of a formal set is established. The planarity rule is used to determine the genera of more general graphs. As an example, genera of a type of 3-connected cubic graphs are provided. In addition, the genus of a graph can be applied in field of moduli spaces of curves.

Date received: November 25, 2015


A (mostly) combinatorial proof of the homology cobordism classification of lens spaces
by
Stephan Wehrli
Syracuse University
Coauthors: Margaret Doig

It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cobordant exactly when they are oriented homeomorphic. We provide a new and mostly combinatorial proof using the Heegaard Floer d-invariants, which themselves may be defined combinatorially for lens spaces.

Date received: November 17, 2015


Knot Theory and some Invariants
by
Ansgar Wenzel
University of Sussex

I am going to give an introduction to General Knot Theory as well as biquandles. In particular, I am going to consider linear and quadratic biquandle functions (i.e. where fa, fb are linear or quadratic polynomials in two variables) and show that there is no commutative quadratic biquandle with quadratic coefficients. Finally, I aim to present a computer programme to calculate quandle and biquandle homology.

Date received: November 18, 2015


The Kauffman Bracket of Infinite Braids
by
Michael Willis
University of Virginia
Coauthors: Gabriel Islambouli Slava Krushkal

It has been known for some time that the (unnormalized) Kauffman bracket of an `infinite torus braid' stabilizes to give a power series representation of the Jones-Wenzl Projector in the relevant Temperley-Lieb Algebra. I will discuss a generalization of this result to iterated limits of other braids.

Date received: October 27, 2015


Copyright © 2015 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.