Topology Atlas | Conferences


Knots in Washington XLVI: 70th Birthday of Oleg Viro;
May 4-6, 2018
George Washington University,
Washington, DC, USA

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

Conference Homepage


Abstracts

On the Kauffman bracket skein algebra of the thickened T-shirt
by
Rhea Palak Bakshi
The George Washington University
Coauthors: Sujoy Mukherjee, Józef Przytycki, Marithania Silvero, Xiao Wang

Skein modules were introduced by Józef Przytycki as a way to extend the knot polynomials of the 1980’s to arbitrary 3-manifolds. In 1997, Charles Frohman and Razvan Gelca established a delightful product-to-sum formula for the multiplication of two curves in the Kauffman bracket skein algebra of the torus times an interval. We try to do the same for the case of the sphere with four holes times the interval (the thickened T-shirt) and present our results to this end. In particular we address the Witten positivity conjecture for the skein algebra of F0, 4. This is joint work with Sujoy Mukherjee, Józef Przytycki, Marithania Silvero and Xiao Wang.

Date received: April 30, 2018


Real flexible knots and real symplectic knots.
by
Johan Björklund
University of Gävle

In this talk we will define flexible knots, objects meant to capture the topological properties of real algebraic knots, and then use them to introduce flexible isotopy, that is, an isotopy which is at all times a flexible knot.

Flexible knots (and links) are less rigid than real algebraic knots. This lack of rigidity allows a complete classification (up to flexible isotopy) utilizing some basic invariants together with Viro's encomplexed writhe.

Since we have a complete classification on the flexible level, any new rigid isotopy invariant for real algebraic knots must then utilize more of the algebraic geometric structure and not use the smooth topology.

If time allows we will discuss real symplectic knots, which have more structure, and so are closer to the real algebraic knots. This is work in progress with Georgios Dimitroglou-Rizell.

Date received: April 23, 2018


Colored Tri-Plane Diagrams and Branched Covers of the Four-Sphere
by
Patricia Cahn
Smith College
Coauthors: Ryan Blair and Alexandra Kjuchukova

Which four-manifolds arise as 3-fold covers of the four-sphere, branched over surfaces with singularities? We study this question using trisections of four-manifolds and colored tri-plane diagrams, and construct large families of such covers.

Date received: April 19, 2018


Torsion in the Khovanov homology of 3-strand torus links
by
Alex Chandler
North Carolina State University
Coauthors: Adam Lowrance, Vassar College; Radmila Sazdanovic, North Carolina State University; Victor Summers, North Carolina State University

In 2012, Shumakovitch showed that homologically thin links have only Z2-torsion in Khovanov homology. This was accomplished by utilizing a connection between the Bockstein and Turner spectral sequences. We show that similar ideas can be used to prove that 3-strand torus links have only Z2-torsion in Khovanov homology.

Date received: April 29, 2018


Obstructions of algebraic Gordian distance one
by
Jie Chen
Tohoku Univ. (until Mar.27), McMaster Univ. (from Sep.7)

The Gordian distance of two knots is defined to be the minimal number of crossing changes from one knot to the other. Based on matrix operations analogous to crossing changes, Murakami introduced the algebraic Gordian distance between two Seifert matrices. We calculate the Blanchfield pairing when the algebraic Gordian distance is one and then use it to improve a result of Kawauchi. We extend this method to torus knots and conclude some new obstructions of algebraic Gordian distance one.

Date received: March 30, 2018


Virtual Seifert Surfaces
by
Micah Chrisman
Monmouth University

A virtual Seifert surface is a planar representation of a Seifert surface of a homologically trivial knot in a thickened surface Σ×[0, 1], where Σ is compact and oriented. The boundary of a virtual Seifert surface is a virtual knot. We present an algorithm for constructing them from a Gauss diagram of the virtual knot and show how they can be manipulated in the plane in a manner that is analogous to classical Seifert surfaces. Some applications are given. Firstly, virtual Seifert surfaces can be applied to the compute slice obstructions of virtual knots, such as directed signatures of almost classical knots. Secondly, the virtual Seifert surface algorithm gives rise to the notion of virtual canonical 3-genus of almost classical knots. This is compared to related notions in the literature, such as the slice genus, the virtual 3-genus, and the canonical genus of Stoimenow-Tchernov-Vdovina.

Date received: April 2, 2018


Hall Algebras of Surfaces
by
Ben Cooper
University of Iowa
Coauthors: P. Samuelson, M. I. Bakhira, M. Messmore

A short survey of recent work on the Hall algebras of surfaces. Relation to quantum topology and the categorification program will be emphasized.

Date received: April 24, 2018


Relationships between quantum and Heegaard Floer Invariants
by
Nathan Dowlin
Columbia University
Coauthors: Akram Alishahi

There are two main conjectures relating Khovanov-type invariants with knot Floer homology. The first is the existence of a spectral sequence from Khovanov homology to delta-graded knot Floer homology, and the second is the existence of a spectral sequence from HOMFLY-PT homology to (bigraded) knot Floer homology. We will define a family of invariants on the knot Floer side which are analogous to the sl(n) homology of Khovanov and Rozansky which shed some light on these conjectures. In the n=2 case, we construct an algebraically defined filtered homology theory such that the E2 page is isomorphic to Khovanov homology, and the higher pages are link invariants. The construction is inspired by counting holomorphic discs in a particular Heegaard diagram, so we expect the E page of the spectral sequence to recover delta-graded knot Floer homology.

Date received: March 30, 2018


Basic moves of rectangular diagrams of surfaces
by
Ivan Dynnikov, Maxim Prasolov
Moscow State University
Coauthors: Ivan Dynnikov

We introduce a simple set of moves that allow for transition between two rectangular diagrams of surfaces if and only if the corresponding surfaces are isotopic. This approach extends to Giroux's convex surfaces.

Date received: April 23, 2018


Minimizing intersections points of flat virtual links
by
David Freund
Dartmouth College
Coauthors: Vladimir Chernov, Rustam Sadykov

A virtual n-string is a collection of n closed curves on an oriented surface M and counting the minimal number of intersection points in the homotopy class of this collection is a classical problem. We address the analogous problem for flat virtual links, i.e., equivalence classes of virtual n-strings related by homotopy and by stabilization/destabilization of the supporting surface. In particular, we use generalizations of the Cahn cobracket and the Andersen-Mattes-Reshetikhin bracket to obtain the minimal number of intersection points for a flat virtual link and show that this value is realized on a minimal genus representative.

Date received: April 20, 2018


Fully classifying virtual string links up to cobordisms
by
Robin Gaudreau
University of Toronto

Virtual string links are oriented tangles with classical and virtual crossings, and such that the endpoints can be grouped into two connected sets, one outgoing and one incoming. While classical knots can all be unknotted using cobordisms, cobordisms classes of virtual string links on n components form a non-trivial free abelian group.

Date received: April 13, 2018


Hilbert schemes and y-ification of Khovanov-Rozansky homology
by
Eugene Gorsky
University of California, Davis
Coauthors: Matthew Hogancamp (USC)

We define and study a deformation of the triply-graded Khovanov-Rozansky homology for links,

with one deformation parameter per each connected component. We explicitly compute

both deformed and undeformed homology for all positive powers of the full twist, and

relate the answer to algebraic geometry of Hilbert schemes of points.

Date received: March 25, 2018


Finite real algebraic curves
by
Ilia Itenberg
Sorbonne Université, Paris

The talk is devoted to real plane algebraic curves with finitely many real points.

We study the following question: what is the maximal possible number of real points

of such a curve provided that it has given (even) degree and given geometric genus?

We obtain a complete answer in the case where the degree is sufficiently large

with respect to the genus, and prove certain lower and upper bounds for the number in question

in the general case.

This is a joint work with E. Brugallé, A. Degtyarev and F. Mangolte.

Date received: April 20, 2018


An Euler-type theorem for the 2-complex of cubes
by
Paul Kainen
Georgetown University

It is shown that the 2-skeleton S of a d-dimensional cube (d ≥ 3 odd) is the union of a finite family of face-disjoint 2-complexes where each is a ball or torus. Further, there is a connected sum basis for S. This is part of ongoing work with Richard Hammack.

Date received: March 27, 2018


Virtual Link Cobordism
by
Louis H Kauffman
University of Illinois at Chicago

This talk will discuss properties of virtual link cobordism, including results of Dye, Kaestner

and Kauffman using Khovanov Homology for virtual links and also using the concordance

invariance of the affine index polynomial.

Date received: March 16, 2018


Little variations on the theme of counting real lines.
by
Viatcheslav Kharlamov
Strasbourg University

As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. In this invariant signed count the input of a line is given by its local contribution to the Euler number of a certain auxiliary universal vector bundle.

The aim of the talk is to present other, in a sense more geometric, interpretations of the signs involved in the invariant count. In particular, this provides some generalizations of Segre indices of real lines on cubic surfaces and Welschinger weights of real lines on quintic threefolds.

This is a joint work with S.Finashin.

Date received: April 22, 2018


Link homology: matrix factorizations and foam evaluations
by
Mikhail Khovanov
Columbia University

We will review and discuss various definitions and constructions of link homology

via matrix factorizations, bimodules and foams, ending with the recent work of

Robert and Wagner on sl(n) foam evaluation.

Date received: April 28, 2018


Stern-Brocot-like trees, continued fractions, and topology
by
Jerzy Kocik
Souther Illinois University (SIUC)

Some intriguing connections between various organizations of rational numbers, including Stern-Brocot tree, number games, and spinor four-fold covering of the Apollonian disk packing will be discussed.

Date received: April 23, 2018


Lifting branched covers to braided embeddings
by
Sudipta Kolay
Georgia Tech

An embedding of a manifold M^k in a trivial disc bundle over N^k is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. One could ask which branched covers lift to braided embeddings. This question has been well studied for honest covering maps by Hansen and Petersen. In this talk, we will discuss this question for branched covers over low dimensional spheres.

Date received: April 5, 2018


Engel relations and 4-manifolds
by
Slava Krushkal
University of Virginia

I will discuss geometric classification techniques in the theory of topological 4-manifolds, surgery and the s-cobordism theorem, which are known to hold for a certain restricted class of fundamental groups, and are open in general. Starting with an introduction to the 4-dimensional topological surgery conjecture, this talk will focus on recent results on the construction of new universal surgery models. The construction relies on geometric applications of the group-theoretic 2-Engel relation. (Joint work with Michael Freedman)

Date received: April 29, 2018


The Walks Model of the Color Jones Polynomial and some Applications
by
Jesse S. F. Levitt
University of Southern California
Coauthors: Nicolle E. S. González, University of Southern California; Mustafa Hajij, University of Southern Florida

The colored Jones polynomial is a quantum knot invariant that plays a central role in low dimensional topology. We review a walks along a braid model of the colored Jones polynomial that was refined by Armond from the work of Huynh and Lê. The walk model gives rise to ordered words in a q-Weyl algebra. We will discuss the computation of this invariant with applications to the Jones unknot conjecture as well as several limiting behaviors with applications to the calculation of the Mahler measure of a knot.

Date received: March 19, 2018


Quantum Knots Revisited
by
Samuel Lomonaco
University of Maryland Baltimore County (UMBC)

We will continue our discussion on quantum knots

Date received: April 18, 2018


Are almost alternating links semi-adequate?
by
Adam Lowrance
Vassar College

An almost alternating link is a non-alternating link with a diagram that can be transformed into an alternating diagram via a single crossing change. A Kauffman state is called adequate if no two arcs in the resolution of the same crossing lie on the same component of the state. A link is semi-adequate if it has a diagram where either the all-A state or the all-B state is adequate. In this talk, we discuss the similarities between the Jones polynomial and Khovanov homology of almost alternating and semi-adequate links.

Date received: April 7, 2018


On odd torsion in even Khovanov homology
by
Sujoy Mukherjee

Khovanov homology, an invariant of knots and links, is a categorification of the Jones polynomial. Z2 torsion in the Khovanov homology of knots and links is very common. In this talk, I will first discuss the recent developments regarding non-Z2 torsion in Khovanov homology. In particular, I will provide examples of links with large even torsion and counterexamples to parts of the PS braid conjecture. I will conclude with examples of knots and links with large odd torsion (e. g. Z27) in their Khovanov homology. Part of this is joint work with Przytycki, Silvero, Wang, and Yang.

Date received: April 30, 2018


2-Verma modules and the Khovanov-Rozansky link homologies, II
by
Grégoire Naisse
Université catholique de Louvain (Belgium)
Coauthors: Pedro Vaz

I'll explain the categorification of parabolic Verma modules for gl(2k) which uses categories of dg-modules over enhanced KLR algebras. In particular, I'll explain how one can interpret usual cyclotomic KLR algebras as minimal models of richer dg-algebras. This extra structure is a key ingredient in the proof of the Dunfield--Gukov--Rassmussen conjecture, stating that the KR gl(N)-link invariant is the homology of the KR HOMFLYPT invariant w.r.t. a certain differential. I'll give the main ideas of the proof.

Date received: February 12, 2018


Trace Digarams and Biquandle Brackets
by
Sam Nelson
Claremont McKenna College
Coauthors: Natsumi Oyamaguchi

Biquandle brackets are skein invariants of biquandle-colored knot and link diagrams. While originally defined using a state-sum formulation, for hand computation it is desirable to have a definition of these invariants via a recursive skein expansion, but there is a problem -- smoothings break the biquandle coloring. In this talk we show how to resolve this issue using signed trace diagrams. We find conditions on biquandle brackets to allow over- and under-pass trace moves and identify a Homflypt-style skein relation at monochromatic crossings. This is joint work with Natsumi Oyamaguchi (Shumei University, Japan).

Date received: March 24, 2018


On real algebraic knots and links in RP^3
by
Stepan Orevkov
University of Toulouse, France
Coauthors: Grigory Mikhalkin

I will give a survey of results about the topology of nonsingular real algebraic curves in RP^3: from the seminal paper of Oleg Viro to the recent results by Mikhalkin and myself.

Date received: April 21, 2018


Looking at Linking Numbers
by
Ken Perko
uninstitutionalized

This paper looks at some low-hanging fruit on the great tree of knot theory. Continuing work begun in 1964, it examines crossing reversals of 3-colored knot diagrams that preserve, or predictably alter, the linking number between branch curves of 3-fold non-cyclic covering spaces. See “Symmetric Quotients of Knot Groups and a Filtration of the Gordian Graph” by S. Baader and A. Kjuchukova at arXix:1711.08144v1 (22 Nov 2017) but don't believe the conjecture at the bottom of page 8.

Date received: January 17, 2018


Index polynomials for virtual tangles
by
Nicolas Petit
Oxford College of Emory University

The Henrich-Turaev polynomial is an index polynomial, and one of the simplest examples of Vassiliev invariants of order one for virtual knots.

We will discuss various generalizations of this polynomial to the case of virtual tangles; these generalizations also happen to be Vassiliev invariants of virtual tangles.

Date received: March 26, 2018


Upper bounds on Virtual Bridge Numbers
by
Puttipong Pongtanapaisan
University of Iowa

The virtual bridge number of a knot is the smallest number of overbridges taken over all virtual knot diagrams of the knot. A naive upper bound obtained from counting the number of overbridges of a virtual knot diagram representing the knot can be much larger than the actual virtual bridge number. In this talk, I will define the Wirtinger number of a knot, which is the minimum number of generators of the knot group over all meridional presentations in which every relation is an iterated Wirtinger relation. The Wirtinger number turns out to be equal to the virtual bridge number, and we will see that the Wirtinger number of a diagram representing the knot gives a stronger upper bound on the virtual bridge number of the knot.

Date received: April 4, 2018


Exchange classes of rectangular diagrams, Legendrian knots, and the knot symmetry group
by
Maxim Prasolov
Moscow State University
Coauthors: Ivan Dynnikov (Steklov Mathematical Institute)

Rectangular diagrams are a particularly nice way to represent knots and links in the three-space. The crucial property of this presentation is the existence of a monotonic simplification algorithm for recognizing the unknot [I.D., 2006]. The present research is motivated by an attempt to extend the monotonic simplification procedure to arbitrary knot types.

Another nice feature of rectangular diagrams is their relation to Legendrian knots. Namely, each rectangular diagram defines, in a very natural way, two Legendrian knots, one with respect to the standard contact structure, and the other with respect to the mirror image of the standard contact structure. These two Legendrian knots always have an important mutual independence property [I.D., M.Prasolov, 2013], which is roughly this: any Legendrian stabilization and destabilization of each of the two Legendrian types can be done without altering the other, by applying elementary moves to the rectangular diagram.

Among elementary moves defined for rectangular diagrams, there are those that preserve both Legendrian knot types associated with the diagram. These are exchange moves. An exchange class is a set of rectangular diagrams that can be obtained from a fixed diagram by exchange moves.

Let K be a topological knot type, and let L1 (respectively, L2) be a ξ+-Legendrian (respectively, ξ--Legendrian) knot type of topological type K, where ξ+ and ξ- are the standard contact structure and its mirror image, respectively. There are symmetry groups G, H1, H2 naturally associated with K, L1, and L2, respectively. We show that the set of exchange classes representing L1 and L2 simultaneously, is in one-to-one correspondence with the set H1\G/H2 of double cosests.

The proof uses, among other things, a trick from a joint work of I.Dynnikov and V.Shastin (in preparation).

Date received: April 22, 2018


Learning from the masters: four lessons from Oleg
by
Jozef H. Przytycki
George Washington University,

(1) Pachner moves: At the beginning of his stay in Riverside, in Spring of 1992, Oleg brought a news about Pachner moves (just add an (n+1)-simplex to the boundary of triangulated (n+1)-manifold (M=∂W). A modification of a triangulation of M is a Pachner move).

(2) Khovanov homology: Oleg gave "Jankowski talk" in Gdansk in 2002, and at seminar after explained us his understanding of Khovanov homology. It was enlightenment for us. Being a skein module person I started thinking on categorification of links in F ×[0, 1].

(3) Finite topological spaces: Oleg gave Colloquium at GWU (2007) and started with a puzzle - compute fundamental group of the following 4-element topological space (X, T). X={a, b, c, d}, T={∅, {b}, {c}, {b, c}, {a, b, c}, {b, c, d}, X}. We looked with surprise: 4-element topological space with interesting fundamental group? At the end of the lecture Oleg commented `if you got Z, it is good'. From that time I devote part of my General Topology class on finite topological spaces. I even noticed that their topology always comes from some hemimetric.

(4) Vladimir A Rochlin: This is most important, of moral character, and concerns Oleg's advisor. Rochlin was dismissed from his job suddenly... But Oleg should tell us this story.

Date received: May 1, 2018


An Application of Salem Numbers to Braid Group Representations
by
Nancy Scherich
UC Santa Barbara working with Darren Long
Coauthors: none

Many well known braid group representations have a parameter. I will show how to carefully choose evaluations of the parameter to force the representation to be discrete, and sometimes even land in a lattice. It is surprising and exciting to see how careful algebraic constructions can lead to geometric results.

Date received: April 9, 2018


On the Jones polynomial via Khovanov and chromatic homology
by
Dan Scofield
North Carolina State University
Coauthors: Radmila Sazdanovic

We compute torsion in Khovanov homology for certain classes of links, using a related homology theory for graphs. We show that for links whose Khovanov homology is thin, the first/last n coefficients of the Jones polynomial are given by a formula in terms of the all-A state graph of a diagram.

Date received: April 29, 2018


Torsion in the Khovanov Homology of Closed 3-Braids
by
Victor Summers
North Carolina State University
Coauthors: Alex Chandler, Adam Lowrance, Radmila Sazdanovic

Integral Khovanov homology is a bigraded homology theory for links in the 3-sphere.

These homology groups may contain torsion subgroups, and efforts have been made

to identify different types of torsion occurring in Khovanov homology for various classes

of knots and links. For example, in 2004 A. Shumakovitch showed that only Z/2-torsion

can appear in the Khovanov homology of non-split alternating links. In this talk I will

use a classification of 3-braids due to Kunio Murasugi, in conjunction with the skein

long exact sequence, to demonstrate that no odd torsion occurs in the Khovanov

homology of closed 3-braids.

Date received: April 30, 2018


2-Verma modules and the Khovanov-Rozansky link homologies, I
by
Pedro Vaz
Université catholique de Louvain
Coauthors: Grégoire Naisse (Univ. catholique de Louvain

In this talk I will explain how to obtain the HOMFLYPT and the gl(N)-link polynomials using parabolic Verma modules for gl(2k). I will then sketch the theory of categorified Verma modules and use it to give a construction of Khovanov-Rozansky's HOMFLYPT and gl(N)-link homologies.

Date received: February 12, 2018


Links in the projective space
by
Oleg Viro
Stony Brook University

Links in the projective space admit a diagrammatic presentation similar to classical link diagrams. This gives an opportunity to transfer approaches and results on classical links, although many details change under the transition. Links in the projective space come naturally as real algebraic curves. This will be discussed in Orevkov's talk. In my talk, I will concentrate on pure topology.

Date received: April 25, 2018


Enumerating quandles and racks up to isomorphism
by
Petr Vojtěchovský
University of Dennver
Coauthors: Seung Yeop Yang

Quandles and racks play an important role in the theory of knot invariants. Presently, quandles have been classified up to order 9 and racks up to order 8. We improve both bounds to 13, using a representation due to Blackburn. We will also describe key features of the algorithm.

Date received: April 6, 2018


On Alexander polynomials of graphs
by
Zhongtao Wu
The Chinese University of Hong Kong
Coauthors: Yuanyuan Bao

Using Alexander modules, one can define a polynomial invariant for a certain class of graphs with a balanced coloring. We will give different interpretations of this polynomial by Kauffman state formula and MOY relations. Moreover, we show it coincides with Viro's gl(1, 1)-Alexander polynomial of a graph.

Date received: January 14, 2018


Homology theory of non-degenerate involutive set-theoretic solutions to the Yang-Baxter equation
by
Seung Yeop Yang
University of Denver
Coauthors: Marco Bonatto, Michael K. Kinyon, David Stanovský, Petr Vojtěchovský

Wolfgang Rump showed that there is a one-to-one correspondence between non-degenerate involutive solutions to the Yang-Baxter equation and bijective rumples. In 2004, a homology theory for set-theoretic Yang-Baxter equations was introduced by Carter, Elhamdadi, and Saito. In this talk, we construct a set-theoretic Yang-Baxter (co)homology theory for bijective rumples and define rumple cocycle invariants of links.

Date received: April 27, 2018


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