Topology Atlas | Conferences

Knots in Washington XXXVII
January 19-20, 2014
George Washington University
Washington, DC, USA

Mieczyslaw K. Dabkowski (UT Dallas), Valentina Harizanov (GWU), Jozef H. Przytycki (GWU and UMCP), Yongwu Rong (GWU), Radmila Sazdanovic (NCSU), Alexander Shumakovitch (GWU), Hao Wu (GWU)

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A filtration on HOMFLY-PT homology via virtual crossings
Michael Abel
University of North Carolina, Chapel Hill
Coauthors: Lev Rozansky

In 2006, Khovanov gave a construction of HOMFLY-PT homology in the homotopy category of Soergel bimodules. Soergel bimodules can be naturally filtered by bimodules representing virtual crossings, known as standard bimodules. We show that, by choosing the proper filtrations by virtual crossings, we get a filtration on HOMFLY-PT homology which is a link invariant. The grading induced by this filtration is independent of the three preexisting gradings, turning HOMFLY-PT homology into a quadruply-graded theory.

Date received: November 2, 2013

The Kauffman bracket ideal for genus-1 tangles
Susan M. Abernathy
Louisiana State University

A genus-1 tangle is a 1-manifold with two boundary components properly embedded in the solid torus. A genus-1 tangle G embeds in a link L if we can complete G to L via a 1-manifold in the complement of the solid torus containing G. A natural question to ask is: given a tangle G and a link L, how can we tell if G embeds in L? We define the Kauffman bracket ideal, which gives an obstruction to tangle embedding, and outline a method for computing a finite list of generators for this ideal. We also give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal and discuss how the concept of partial closures relates to this ideal.

Date received: November 30, 2013

A Geometric Description of Turaev Surfaces
Cody Armond
University of Iowa
Coauthors: Nathan Druivenga, Thomas Kindred

Turaev surfaces are surfaces constructed from a knot diagram, which were originally studied to help solve the problem of whether reduced alternating diagrams have minimal crossing number. They are constructed from two opposing state arising in the state sum formula of the Kauffman bracket polynomial. Since then, they have been further studied and have been shown to have relationships with the rank polynomial for ribbon graphs as well as knot Floer homology and Khovanov homology.

We present a description of Turaev surfaces which is not dependent on knot diagrams, and is instead described as a Heegard splitting of S3 with special properties.

Date received: December 15, 2013

Knot contact homology and a question of Cappell and Shaneson
Christopher Cornwell
Duke University
Coauthors: David Hemminger

An open question raised by Cappell and Shaneson asks whether the minimal number of meridians needed to generate the group of a knot is equal to the bridge number of that knot. Knot contact homology gives a new approach to studying this question, which we will show behaves well for certain satellites. This gives an affirmative answer to Cappell and Shaneson's question for many satellite knots, including iterated torus knots.

Date received: January 3, 2014

Properties of Cohesive Powers
Rumen Dimitrov
Department of Mathematics, Western Illinois University, Macomb IL 61455

The cohesive sets are infinite sets that cannot be split into two infinite subsets by computably enumerable sets. Each cohesive set can be used to define an ultrafilter in the Boolean algebra of computable sets. We will introduce the notion of cohesive power ∏C\QTRcalA of a computable structure \QTRcalA over a cohesive set C. We will give examples of algebraic structures \QTRcalA such that \QTRcalA and ∏C\QTRcalA are isomorphic as well as when \QTRcalA and ∏C\QTRcalA do not satisfy the same first order formulas. We will discuss the question of isomorphisms of ∏C1\QTRcalA and ∏C2\QTRcalA when C1 and C2 are cohesive.

Date received: December 19, 2013

Torus knots and rational Cherednik algebras
Eugene Gorsky
Columbia University
Coauthors: Pavel Etingof, Ivan Losev

We prove that all colored HOMFLY-PT invariants of torus knots have nonnegative coefficients

when expanded in (-a) and q. Moreover, these coefficients are equal to the dimensions of certain

components in certain irreducible representations of rational Cherednik algebras. The talk is based

on arXiv:1304.3412.

Date received: December 27, 2013

The colored Kauffman skein relation and the tail of the colored Jones polynomial
Mustafa Hajij
Louisiana State University

We use the colored Kauffman skein relation to give a new simple proof for the stability of the coefficients of the colored Jones polynomial of alternating links.

Date received: October 30, 2013

The left-orderability and the cyclic branched coverings
Ying Hu
Louisiana State University

We give a sufficient condition for the fundamental group of the nth cyclic branched covering of S3 along a prime knot K to be left-orderable in terms of representations of the knot group. As an application, we show that for any (p, q) two-bridge knot, with p ≡ 3 mod 4, there are only finitely many cyclic branched coverings whose fundamental groups are not left-orderable. This answers a question posed by D abkowski, Przytycki and Togha.

Date received: November 9, 2013

Milnor-Turaev torsion and Iwasawa theory
Jonathan Huang
University of Maryland, College Park

There is an analogy between primes and knots, stemming from the fact that the etale fundamental group of a finite field is the profinite completion of the integers. Thus, a prime number sitting inside Spec Z behaves like a circle S1 sitting inside the sphere S3. After giving an introduction to the basic features of arithmetic topology, we will focus on the similarities between torsion invariants of Milnor and Turaev and the main conjecture in Iwasawa theory. The Alexander polynomial is analogous to the Iwasawa polynomial, and Milnor-Turaev torsion is analogous to the 'zeta element' of Kato. Finally, we will briefly discuss questions arising from refined versions of the Iwasawa main conjecture involving higher Fitting ideals.

Date received: December 22, 2013

Comparing the slice and ribbon genera of knots via braided surfaces
Mark Hughes
Stony Brook University

In this talk I will discuss bounds on the slice genus of a knot coming from it's representation as a braid closure, starting with the slice-Bennequin inequality. From there I will use surface braiding techniques of Rudolph and Kamada to exhibit a new lower bound on the ribbon genus of a knot, given some knowledge about what slice surfaces it bounds.

Date received: January 17, 2014

A folding technique for drawing circulant graphs in books
Paul C. Kainen
Georgetown University

A folding technique for laying out the vertices of some circulants in an outerplanar drawing makes it easy to find good drawings of these graphs in books with low book thickness and crossing number. Let C(n, {1, k}) denote the circulant with edges of length 1 and k, while crk(G) is the k-page crossing number of G in a book with k pages.

Theorem 1. For k ≥ 2 an integer and r ≥ 2 even, the book thickness of C(rk, {1, k}) is at most 4.

Theorem 2. For k ≥ 3 odd and r ≥ 2 even, cr2(C(rk, {1, k}) ≤ r(k-2) and cr3(C(rk, {1, k}) ≤ r/2.

The folding technique achieving these results is described for the case r = 2 in the author's technical report, Circular layouts for crossing-free matchings,

Date received: January 8, 2014

Stable Concordance Genus of Knots
Kate Kearney
Louisiana State University

The concordance genus of a knot is the least three–genus of a knot concordant to the knot. The concordance genus is bounded below by the four–genus (or slice genus), and bounded above by the three–genus. This makes the concordance genus a valuable tool to describe the difference between these invariants. In simple cases the concordance genus is not difficult to calculate, since there are a variety of algebraic tools that give bounds for the concordance genus. Unfortunately, as the crossing number increases, it becomes increasingly difficult to find concordances. The stable concordance genus, which we will discuss in this talk, describes the behavior of the concordance genus of a given knot under connect sum. We will briefly define the invariant, give some examples of calculations, and discuss applications to the study of concordance.

Date received: December 11, 2013

Relation between the Milnor's bar-mu-invariant and HOMFLYPT polynomial
Yuka Kotorii
Tokyo Institute of Technology
Coauthors: Akira Yasuhara

For an ordered, oriented link in the 3-sphere, J. Milnor defined a family of invariants, known as Milnor bar-mu-invariants. We show that any Milnor bar-mu-invariant of length between 4 and 2k + 2 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all Milnor invariants of length ≤ k vanish. In particular, for any 4-component link the Milnor bar-mu-invariants of length 4 can be represented as a combination of HOMFLYPT polynomial of knots. This is a joint work with Akira Yasuhara.

Date received: December 29, 2013

A colored operad for string link infection
Robin Koytcheff
University of Victoria
Coauthors: John Burke

Budney recently constructed an operad which encodes splicing of knots and gave a decomposition of the space of knots over this splicing operad. Infection of links by string links is a generalization of splicing from knots to links and is useful in studying knot concordance. We construct a colored operad that encodes infection. This colored operad captures all the relations in the 2-string link monoid. We also show that a certain subspace of 2-string links is freely generated over a suboperad of our infection colored operad by its subspace of prime links.

Date received: December 18, 2013

Qualgebras and knotted 3-valent graphs
Victoria Lebed
Osaka City University
Coauthors: Seiichi Kamada

The quandle structure can be seen on the one hand as an algebraic counterpart of knots, and on the other as an axiomatic expression of the properties of conjugation in a group (where one forgets the multiplication and the inverse operations). In this talk we introduce the notion of qualgebra (= quandle + algebra), which is a quandle endowed with an additional "multiplication" operation, compatible with the quandle operation in a certain way. It can be seen on the one hand as an algebraic counterpart of 3-valent knotted graphs, and on the other as an axiomatic expression of the properties of conjugation and multiplication in a group (where one forgets the inverse operation). We discuss the properties of qualgebras, various examples (including those not coming from groups), and associated invariants of 3-valent graphs.

Date received: December 6, 2013

The Khovanov-Rozansky concordance invariants
Lukas Lewark
Durham University, UK
Coauthors: Andrew Lobb

The Khovanov-Rozansky homologies induce a family of knot concordance invariants

that give strong lower bounds to the slice genus,

and wedge themselves between the smooth and the topological category.

First of these Khovanov-Rozansky concordance invariants is the Rasmussen invariant,

but we will see many others, which can be distinguished using spectral sequences.

Geometrical applications will be discussed.

Date received: January 7, 2014

On finite-type invariants of virtual knots.
Arnaud Mortier
Osaka City Advanced Mathematical Institute

Gauss diagrams are a combinatorial presentation of knot diagrams, at the origin of the virtual knot theory due to Kauffman. Goussarov-Polyak-Viro and Fiedler showed that those diagrams are very convenient to describe finite-type invariants. We give a positive answer to a conjecture due to Polyak on the structure of these invariants in the case of virtual knots.

Date received: November 1, 2013

Links with binary relations
Maciej Niebrzydowski

We define homology of binary relations, and merge it with quandle homology, to obtain homology for partial quandles with binary relations. It can be used for analyzing link diagrams with binary relations on the set of components with the condition that a component C_2 can move over a component C_1 if C_1 R C_2, where R is a relation. We also discuss some other conditional theories.

Date received: January 9, 2014

Nonorientable genus of a knot in punctued CP2
Kouki Sato
Tokyo Gakugei University
Coauthors: Motoo Tange (University of Tsukuba)

Let K be a knot in ∂(CP2 - Int B4) and F ⊂ CP2 - Int B4 a smoothly embedded nonorientable surface with boundary K. We prove that if F represents zero in H2( CP2 - Int B4, ∂( CP2 - Int B4); Z2), then β1(F) ≥ - σ(K)/2 + d(S31(K)) - 1, where d(S31(K)) is the Heegaard Floer d-invariant of the integer homology sphere given by 1 surgery on K. In particular, if K is #n 942, then the minimal first Betti number of such surfaces is equal to n - 1.

Date received: November 26, 2013

Effective 9-colorings for knots
Shin Satoh
Kobe University
Coauthors: Yasutaka Nakanishi and Takuji Nakamura

For a composite odd integer n¥geq 9, we introduce the notion of an effective n-coloring for a 1- or 2-dimensional knots and give a lower bound for the minimal number of colors for all effective n-colorings. In particular, we prove that any effectively 9-colorable 1- or 2-knot is presented by some diagram where exactly five colors of nine are assigned to the arcs or sheets.

Date received: January 7, 2014

Unification of associative, self-distributive, and Lie algebras.
Adam S. Sikora
University at Buffalo, SUNY

We define a new class of rings which unifies associative rings, self-distributive rings, and Lie algebras. We will discuss their fundamental properties and their cohomology theory unifying cohomology theories of these different types of algebras.

Date received: November 22, 2013

Lagrangian cobordisms and pseudoisotopy
Lara Simone Suárez
Université de Montréal

Lagrangian submanifolds are central objects in the study of symplectic manifolds. Given two Lagrangian submanifolds L0, L1 in the symplectic manifold (M, ω), a Lagrangian cobordism between them is a cobordism
(W; L0, L1), that can be embedded as a Lagrangian submanifold in (([0, 1]×R) ×M, dx∧dy ⊕ω), with the property that near the boundary it looks like the products [0, ε)×{1} ×L0 and (1-ε, 1] ×{1} ×L1 for some ε > 0. In recent work Biran and Cornea proposed the following conjecture: Exact Lagrangian cobordism implies pseudoisotopy. In this talk we give partial results towards this conjecture.

Date received: December 16, 2013

Positive Links
Eamonn Tweedy
Rice University
Coauthors: Tim Cochran (Rice)

Cochran and Gompf defined a notion of positivity for concordance classes of knots that simultaneously generalizes the usual notions of sliceness and positivity of knots. This positivity essentially amounts to the knot being slice in a positive-definite simply-connected four manifold. We discuss an analogous property for links, and describe relationships with (generalized) Sato-Levine invariants, Milnor's linking invariants, the Conway polynomial, and some modern invariants.

Date received: October 29, 2013

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