Topology Atlas | Conferences


Knots in Washington XLIII; 60th birthday of J. Scott Carter
December 9-11, 2016
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

Quantum, logic and computing
by
Areski Nait Abdallah
Univ. of Western Ontario, Canada and INRIA Paris, France

We discuss the relevance of formal logic proofs in the formalization of quantum information in quantum mechanics, and their application to some quantum mechanics paradoxes.

To that end we use lambda-calculus and the logic of partial information.

Date received: November 8, 2016


Stable colored HOMFLY-PT homology for torus links
by
Michael Abel
Duke University

Webster and Williamson’s colored HOMFLY-PT homology associates to a link, colored by positive integers, a triply-graded vector space. If the link is colored by a single integer k, then this homology theory categorifies the 1k-colored HOMFLY-PT polynomial. In this talk we will explore the stabilization of colored HOMFLY-PT homology for torus links. We prove that the chain complex of bimodules for an infinite full twist braid stabilizes under all colorings and work out the explicit homology of the closure in a few simple cases.

Date received: December 3, 2016


A family of self-trial ribbon graphs that are not self-dual
by
Lowell Abrams
The George Washington University
Coauthors: Jo Ellis-Monaghan

We present a new framework for studying orbits and stabilizers of the ribbon group action on ribbon graphs. This generalizes the action of the Wilson group, which combines the actions of dualization and Petrialization (adding a twist to each ribbon). We then highlight a new infinite family of self-trial ribbon graphs that are not self-dual. This family has two novel aspects - its members are relatively quite small, and are the first known examples which are not Cayley maps.

Date received: November 18, 2016


Whole genome duplication and embedded graphs
by
Nikita Alexeev
GWU
Coauthors: Pavel Avdeyev and Max A. Alekseyev

Genome median and genome halving are combinatorial optimization problems that aim at reconstructing ancestral genomes by minimizing the number of evolutionary events between them and the genomes of extant species. In the talk we will give a formulation of these comparative genomics problems in terms of embedded graphs. Namely, we introduce the following problem: for a given embedded graph G, find the shortest sequence of surgeries (operations, which cut the surface along two edges of G and glue the resulting four sides in a new order) that results in an graph G' such that it has the maximal possible number of connected components and each of its connected components is embedded into a sphere.

Date received: December 9, 2016


An intrinsic topology on a quandle
by
Yongju Bae
Kyungpook National University, Republic of Korea
Coauthors: Byeorie Kim

Even thouth quandles are defined as algebraic structure, they are closely related topological space such as knot theory. In this talk, we will define an (intrinsic) topology on a quandle which is drived from the connectedness as quandle structure and check various properties of quandles with the intrinsic tpology, as view point of point-set topology.

Date received: November 14, 2016


A Poly-Time Knot Polynomial Via Solvable Approximation
by
Dror Bar-Natan
University of Toronto

Rozansky (1996) and Overbay (2013) described a spectacular knot polynomial that failed to attract the attention it deserved as the first poly-time-computable knot polynomial since Alexander's (1928) and (in my opinion) as the second most likely knot polynomial (after Alexander's) to carry topological information. With Roland van der Veen, I will explain how to compute the Rozansky polynomial using some new commutator-calculus techniques and a Lie algebra g1 which is at the same time solvable and an approximation of the simple Lie algebra sl(2).

More at http://drorbn.net/GWU-1612.

Date received: November 17, 2016


Linking Numbers in 3-Manifolds
by
Patricia Cahn
Smith College
Coauthors: Alexandra Kjuchukova

We give an explicit algorithm for computing linking numbers between curves in an irregular dihedral p-fold branched cover of S3. This work extends a combinatorial algorithm by Perko which computes the linking number between the branch curves in the case p=3. Owing to the fact that every closed oriented three-manifold is a dihedral three-fold branched cover of S3, the algorithm given here can be used to compute linking numbers in any three-manifold, provided that the manifold is presented as a dihedral cover of the sphere.

Date received: November 29, 2016


Locating Boundaries of Machine Learning
by
Wesley Calvert
Southern Illinois University

Definitions for machine learnability are well-established. However, they can be difficult to check in a particular case. Much of the literature seems to consist of an ad-hoc algorithm for learning examples of a particular kind, proving that the respective class is learnable.

The main contribution of this talk is a precise calculation of the difficulty of determining whether a class is learnable or not.

On the other hand, the main technical challenge of that calculation is defining a topological setting sufficiently broad for the calculation to be meaningful, but sufficiently narrow for it to be possible.

Date received: December 2, 2016


Globular: Manipulating knots, knotted surfaces, and higher dimensional knots
by
J. Scott Carter
University of South Alabama
Coauthors: Jamie Vicary

About 7 months ago, Jamie Vicary showed me program globular.science . This is a categorical approach to knot theory and higher dimensional knot theory. In addition, diagrammatic calculations can be made in the context of Frobenius or Hopf algebras or their categorical analogues. In this talk, I want to demonstrate a wide variety of example computations that can be made in the context of globular. In particular, I want to show examples of braided n-manifolds embedded and immersed in (n+2)-space. These are two and three fold simple branched covers that are embedded in such a way that the projection induces the branched covering.

Other interesting examples will also be constructed and demonstrated.

Date received: November 23, 2016


Rack homology group of a certain finite quandle
by
Seonmi Choi
Kyungpook National University, Republic of Korea
Coauthors: Yongju Bae

A quandle is a set equipped with a binary operation satisfying three quandle axioms and a rack is an algebraic structure satisfying only second and third quandle axioms except first one. A finite quandle also can be expressed as a sequence of permutations of the underlying set satisfying certain conditions. In this talk, we will study the second rack homology group of the disjoint union of two finite quandles and the second rack homology group of a finite quandle whose sequence is [1, ..., 1, σ, σi] where σ is a permutation of the underlying set X={1, 2, ..., n}.

Date received: November 15, 2016


Random 2-bridge Chebyshev billiard table diagrams
by
Moshe Cohen
Vassar College
Coauthors: Chaim Even-Zohar (Hebrew University of Jerusalem) and Sunder Ram Krishnan (Technion)

Koseleff and Pecker show that all knots can be parametrized by Chebyshev polynomials in three dimensions. These long knots can be realized as trajectories on billiard table diagrams. We use this knot diagram model to study random knot diagrams by flipping a coin at each 4-valent vertex of the trajectory.

We truncate this model to study 2-bridge knots together with the unknot. We give the exact probability of a knot arising in this model. Furthermore, we give the exact probability of obtaining a knot with crossing number c.

Date received: November 17, 2016


Curve complex cohomology
by
Heather A. Dye
McKendree University

From a curve complex, I construct a (co)homology. The curve complex is

collection of oriented curves with trace marks. Using the trace marks and orientation, we define boundary maps and a homology.

Date received: November 11, 2016


Classifications of Topological Quandles on the reals.
by
Mohamed Elhamdadi
University of South Florida
Coauthors: A. Bouziad and Z. Cheng

The problem of classification of topological quandles will be discussed.

We will give the complete classification of topological Alexander quandles over the real line.

A conjecture about the classification on the closed interval [0,1] will be explained.

Date received: December 1, 2016


Based Matrices for Links
by
David Freund
Dartmouth College

Flat virtual links can be realized as free homotopy classes of loops on surfaces considered up to stabilization and destabilization. In 2004, making use of a particular surface representation, Turaev developed a based matrix for flat virtual knots and discovered several corresponding knot invariants. We construct a generalization of Turaev's based matrix for flat virtual links and discuss analogous link invariants.

Date received: November 10, 2016


Representations of the Kauffman Bracket Skein Algebra at Roots of Unity
by
Charles Frohman
University of Iowa
Coauthors: Joanna Kania-Bartoszynska, Thang Le

We prove the Unicity Conjecture of Bonahon and Wong. Generically irreducible representations of the Kauffman bracket skein algebra at roots of unity are parametrized by the maximal ideals of its center.

Date received: September 7, 2016


Twist Regions and Coefficients Stability of the Colored Jones Polynomial
by
Mustafa Hajij
University of South Florida
Coauthors: Mohamed Elhamdadi, Masahico Saito

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram.

This gives us an infinite family of q-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.

Date received: November 8, 2016


On an algebraic description of marked braid diagrams for surface-links
by
Michal Jablonowski
University of Gdansk

We will discuss a method for presentation of knotted surfaces in the four space by investigating a monoid corresponding to the braid form of marked graph diagrams, where algebraic relations on words will be derived from the topological Yoshikawa moves (which sufficiency was proved by Swenton, Kearton and Kurlin). This method start with the use of transverse cross-sections (by Fox and Milnor) and producing a four-valent graph from the hyperbolic splitting (introduced by Lomonaco, Kawauchi, Shibuya, Suzuki and Kamada) of a knotted surface. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful.

Date received: November 11, 2016


Virtual doodles and a quandle type invariant
by
Naoko Kamada
Nagoya City University
Coauthors: Andrew Bartholomew, Roger Fenn, Seiichi Kamada

A doodle was defined by R. Fenn and P. Tayler in 1979. In their definition, a doodle is an equivalence class of a collection of embedded circles in the 2-sphere by Reidemeister move of type II. M. Khovanov extended it to immersed circles in 2-sphere. We generalize it to immersed circles on surfaces modulo surface surgeries besides Reidemeister moves of type I and II. Then doodles on surfaces correspond to virtual doodles on the plane. We also discuss a quandle type invariant for virtual doodle.

Date received: September 15, 2016


Clasp-ribbon surface-links in 4-space
by
Seiichi Kamada
Osaka City University
Coauthors: Kengo Kawamura

Ribbon surface-links are surface-links which bound handlebodies in 4-space with ribbon singularities. We generalize them to clasp-ribbon surface-links, which bound handlebodies in 4-space with clasp or ribbon singularities. We give a characterization of them, and discuss their normal forms.

Paper reference: arXiv:1602.07855

Date received: September 12, 2016


Stable categories of Hopf algebra modules
by
Mikhail Khovanov
Columbia University

Stable category of modules over a finite-dimensional Hopf algebra is triangulated monoidal. We'll discuss several uses of these categories in homological algebra and categorification.

Date received: December 6, 2016


A topological characterization of toroidally alternating knots
by
Seungwon Kim
The Graduate Center, CUNY

We extend Howie’s characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots which is different from Ito’s recent characterization.

Date received: November 30, 2016


The Jones polynomial of almost-alternating and Turaev genus one links
by
Adam Lowrance
Vassar College

In this talk we show that either the first or last nonzero coefficient of the Jones

polynomial has absolute value one for both almost-alternating and Turaev genus one

links. We also discuss a potential generalization of this result to Khovanov homology.

Date received: November 5, 2016


Braids Groups, Higher G_n^k Groups, and Imaginary Generators
by
Vassily Olegovich Manturov
Bauman Moscow State Technical University
Coauthors: (Partially) Seongjeong Kim

(To J.S.Carter, on the occasion of this 60th birthday.)

Given a braid-word in standard generators of the Artin braid group.

Our goal is to read between letters. Namely, for a given braid-word we are going to construct a word in a larger set of generators such that when deleting new generators, we shall get the initial word and the rule of "splicing" the new letters is algebraically meaningful.

Algebraically, we wish to construct a monomorphism from a smaller group to a larger group such that its composition with the obvious forgetful map (deleting new letters) is the identical map. The new generators (letters) are not visible in the initial work, we call them imaginary generators.

Topologically, this construction comes from the following argument. In 2015, the author defined the two-parametric family of groups Gnk, and formulated the main principle:

If a dynamical system of n moving particles possesses a nice codimension one property governed by some k particles then this dynamical system has an invariant valued in the group Gnk.

The usual Artin presentation is spiritually of Gn2 nature: we can consider a dynamics of points on the line and whenever some of them coincide, we put a generator. When dealing with the genuine braid group, we can think of some n distinct points moving on the plane, and we put a generator of the Artin braid group when some two points have the same x coordinate.

However, there is a more interesting way of looking at dynamics of n points on the plane: we can associate a generator of the group Gn3 with the situation when some three points are collinear. This was done in 2015 in a joint work with I.M.Nikonov.

Now, one can mention that the former Gn2 nature of usual Artin's generators is just a partial case of the new Gn3-approach if we just add some infinite point. Indeed, if add an "infinite" point with coordinates (0, -∞) then the condition that some two points have the same abscissa is exactly the same as the condition that these two points belong to a line passing through (0, ∞). Thus, having a braid-word, we get a word in a larger alphabet (a modification of the Gn+13 presentation) with the initial word inside.

It is well known that the group Gn3 has lots of nice invariants of crossings valued in free products of cyclic groups.

The "pull-back" of this Gn3-approach allows one to construct various invariants of letters in a given Artin braid word in standard generators.

By looking at classical crossings, we can get various consequences on minimal crossing number and unknotting number for classical braids. This is a partial work with Seongjeong Kim.

If I have time, I will say how the higher groups Gnk (with large k) are related to fundamental groups of configuration spaces of higher dimensions.

Where else can we read between letters to study group presentations better? Topology and geometry can lead us to some partial answers to this question.

Date received: December 7, 2016


Dubrovnik and Kauffman two variables skein modules of the lens spaces L(p, 1)
by
Maciej Mroczkowski
University of Gdansk

We present the Dubrovnik and Kauffman skein modules of L(p, 1). W show that the Dubrovnik skein modules are free, as well as the Kauffman skein modules when p is odd. For p even, we show that there is torsion in the Kauffman skein modules.

Date received: November 28, 2016


Biquandle Virtual Brackets
by
Sam Nelson
Claremont McKenna College
Coauthors: Kanako Oshiro (Sophia University), Ayaka Shimizu (Gunma National College of Technology) and Yoshiro Yaguchi (Gunma National College of Technology)

A biquandle bracket is a skein invariant for biquandle-colored knots and links with coefficients depending on the biquandle colors at a crossing. A biquandle virtual bracket adds a virtual crossing interpreted as a kind of smoothing, with coefficients depending of the biquandle colors at each crossing. The enhancements of the biquandle counting invariant determined by biquandle virtual brackets include classical quantum invariants and biquandle cocycle invariants as special cases.

Date received: September 15, 2016


Straight knots - A new invariant
by
Nicholas Owad
Gettysburg College

A knot is said to be straight if all the crossings occur along a straight strand. This yields a new invariant which we call the straight number, which is the minimum number of crossings required with the knot in straight position. We will discuss an algorithm for computing the invariant for the standard table of knots and some relations to other invariants.

Date received: November 2, 2016


Linking in 3-colored knot covers
by
Ken Perko
Uninstitutionalized (Ken of Perko)

This is an extension of our talks at KIW XL and XLII, all in the spirit of Reidemeister's "Anschaung als Erkenntnisquelle" [JFM 61.0969.01 (1935)]. It adds some belated frosting to the cake of our 1964 Princeton undergraduate thesis by showing how a certain type of 3-colored knot diagram may be manipulated to produce different knots with the same linking number between branch curves. Cf. JKTR 25 (2016) 1640010 (8 pages). We like to think of it as topology for fifth graders.

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Date received: November 6, 2016


In the steps of Scott: studying distributive homology
by
Jozef H. Przytycki
George Washington University

I will describe my path to distributive and Yang-Baxter homology, following the path of Scott Carter

Date received: October 30, 2016


Heegaard Floer invariants of knots in the 4-sphere
by
Daniel Ruberman
Brandeis University
Coauthors: Adam Levine

We use a partially twisted version of Heegaard Floer homology to define invariants of 3-manifolds, generalizing the d-invariant of a rational homology sphere. A Seifert surface for an oriented knot in the 4-sphere has a canonical twisting, and its d-invariant turns out to depend only on the original knot.

Date received: September 17, 2016


Knot colorings by quandles and their animations
by
Masahico Saito
University of South Florida
Coauthors: W. Edwin Clark

A Fox coloring of a knot diagram is defined by assigning integers modulo n to arcs of the diagram with a certain condition at every crossing. The number of colorings is independent of choice of a diagram, and is a knot invariant. This idea leads to a concept of an algebraic system called quandles, that have self-distributive binary operations with a few other conditions. Knot colorings are defined with quandles and yield knot invariants. This was further generalized to knot invariants called quandle cocycle invariants, incorporating ideas from quantum knot invariants and group cohomology. After a review of these concepts, we consider quandle cocycle invariants with matrix groups. A continuous family of knot colorings is represented by animations of polygons moving on the sphere. These animations will be presented.

Date received: December 2, 2016


Weakness of two surface-knot theory models
by
Jonathan Schneider
University of Illinois Chicago
Coauthors: Eiji Ogasa, Louis Kauffman

We consider two 2-dimensional diagrammatic knot theories and an invariant model for each. The Rourke model for Rotwelded surface-knot theory is non-complete; that is, there exist distinct Rotwelded surface-knots with equivalent Rourke models. We provide an example of such. Also, the Eiji model for Virtual surface-knot theory is non-complete; we provide an example of this as well.

Date received: November 2, 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: November 14, 2016


The Alexander polynomial of some virtual knots via the multi-variable Alexander polynomial of links
by
Robert Todd
Mount Mercy University
Coauthors: Micah Chrisman

Almost classical virtual knots are those that are homologically trivial in the thickened Carter surface. Boden et al. show these virtual knots have an Alexander polynomial whose definition is analogous to that for classical knots. Using the theory of virtual covers we show that for some almost classical virtual knots their Alexander polynomial can be found as an evaluation of the classical multi-variable Alexander polynomial of a corresponding two component link. We also comment on an interpretation of the index of a crossing in a virtual knot. Several specific examples will be illustrated.

Date received: September 9, 2016


Alexander polynomials of simple-ribbon knots
by
Tatsuya Tsukamoto
Osaka Institute of Technology
Coauthors: Tsuneo Ishikawa, Kengo Kishimoto, Tetsuo Shibuya

An SRm-fusion is a certain m-ribbon fusion and an SRm-knot is a knot obtained from the trivial knot by SRm-fusions, where m is a non-negative integer.

We determine the Alexander polynomials of SRm-knots. For distinct non-negative integers m and n, we also give a necessary condition for knots to be an SRm-knot and an SRn-knot.

Date received: November 20, 2016


K-theory and 1-dimensional supersymmetric Euclidean field theories
by
Peter Ulrickson
The Catholic University of America

We describe the construction of spaces representing K-theory by means of certain functorial quantum field theories in such a way that precomposition of 1-d topological quantum field theories with a functor that forgets geometry induces group completion on classifying spaces of functor categories.

Date received: December 5, 2016


Formal proofs in low-dimensional topology
by
Jamie Vicary
University of Oxford

We introduce a new tool, the proof assistant Globular, for the formalization and development of Morse-theoretic proofs in low-dimensional topology, of the sort that Scott Carter has developed so beautifully throughout his career.

Date received: September 22, 2016


On Milnor's link-homotopy invariants
by
Kodai Wada
Waseda University

J. Milnor defined two kinds of link-homotopy invariants [&175;(μ)] and μ*. By definition it would seem that the [&175;(μ)]-invariant is stronger than the μ*-invariant. The problem was posed at the 1982 Sussex conference if the [&175;(μ)]-invariant is actually stronger. In this talk we give infinitely many examples that for non-repeated sequences of length greater than four [&175;(μ)] and μ* are distinct. For sequences of length not greater than four, Milnor has shown that [&175;(μ)]=μ*.

Date received: September 15, 2016


Khovanov homology of infinite braids
by
Michael Willis
University of Virginia
Coauthors: Gabriel Islambouli

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Time permitting, I will also discuss extensions to more general infinite braids.

Date received: November 3, 2016


Distributive Structure Homology and its Applications to Knot Theory
by
Seung Yeop Yang
George Washington University
Coauthors: J. Scott Carter and Jozef H. Przytycki

The first homology theory using right distributive structures, called rack homology, was introduced by Fenn, Rourke, and Sanderson. Przytycki constructed one-term and multi-term distributive homology theories as generalizations of Fenn, Rourke, and Sanderson's studies. Meanwhile, the rack homology theory was modified to so-called quandle homology by Carter, Jelsovsky, Kamada, Langford, and Saito in order to define cocycle invariants of knots.

We study annihilation of torsion in rack and quandle homology of some finite quandles. Moreover, we discuss applications of distributive structure homology in classical and higher dimensional knot theory.

Date received: December 3, 2016


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