Topology Atlas | Conferences


Knots in Washington XLVIII
May 10-12, 2019
George Washington University
Washington, DC, United States

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

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Abstracts

Gram Determinants Motivated by Knot Theory
by
Rhea Palak Bakshi
The George Washington University
Coauthors: Dionne Ibarra, Sujoy Mukherjee and Józef H. Przytycki

Linking matrices, Seifert matrices and Goeritz matrices are early examples of the use of Gram type determinants in knot theory. In this talk, I will first discuss the Gram determinant of type A which was used by Lickorish to construct the Witten – Reshetikhin – Turaev invariants of 3 - manifolds combinatorially. I will then discuss the Gram determinant of type B, whose closed formula was given by Chen and Przytycki, which answered the question first posed by the late combinatorialist Rodica Simion. Finally, I will describe our work on the generalized Gram determinant of type A and the Gram determinant of type Mb.

Date received: May 3, 2019


The generalized Alexander polynomial and virtually slice knots
by
Micah Chrisman
The Ohio State University
Coauthors: Hans U. Boden

In 1994, Jaeger, Kauffman, and Saluer (JKS) gave a determinant formulation for the Alexander-Conway polynomial. Their polynomial, which was inspired by the free fermion model in statistical mechanics, can be used to define an invariant of knots in thickened surfaces Σ×[0, 1]. Sawollek further showed that the JKS polynomial is an invariant of virtual knots. Silver-Williams then proved that this is equivalent to the generalized Alexander polynomial of virtual knots. In 2008, Turaev defined a new notion of sliceness for knots in thickened surfaces. A knot K ⊂ Σ×[0, 1] is said to be (virtually) slice if there is a compact connected oriented 3-manifold W and a disc D smoothly embedded in W ×[0, 1] such that ∂W=Σ and ∂D=K. Here we show that the generalized Alexander polynomial is vanishing on all virtually slice knots. To do this, we prove that Bar-Natan's "Zh" correspondence and Satoh's Tube map are both functorial under concordance. As an application, we determine the slice status and slice genus of many low crossing virtual knots. Indeed, a knot K ⊂ Σ×[0, 1] is virtually slice if and only if the corresponding virtual knot is slice in the sense of Kauffman. This project is joint work with H. U. Boden (https://arxiv.org/pdf/1903.08737.pdf).

Date received: April 14, 2019


Second quandle homology from Schur multiplier (the historical perspective)
by
Jozef H.Przytycki
George Washington University
Coauthors: Rhea Palak Bakshi, Dionne Ibarra, Sujoy Mukherjee, Takefumi Nosaka

The historical perspective starts form my observation that the second homologies of an odd abelian group coincides with the second homology of assocuated Takasaki quandle of the group (a*b=2b-a).

We express the second quandle homology of a quasigroup Alexander quandles in terms of the exterior algebra of X. We present a short self-contained proof of its structure and provide some computational examples. The result is as follows: Let X be an Alexander quandle with (1-t) invertible. Then there is an isomorphism
H2Q(X, Z) = X∧Z X

(x∧x- tx∧tx)
.
We discuss also and example of connected Alexander quandle which is not a quasigroup:

Cosider the countable direct sum of a group Z indexed by positive numbers:
X=⊕i > 0 Z(i)
with 1i the identity of Z(i)
Let f:X→ X be a epimorphism given on the basis by f(1i)=1i-1 for i > 1 and f(11)=0. Clearly f is not a monomorphism. We have (1-f)(1i) = 1i-1i-1 and observe that (1-f) is invertible with the inverse given by:
(1-f)-1(1i)=1i+1i-1+...+11.
Now observe that if we put t=1-f then (X, *) with a*b=ta +(1-t)b is a quandle which is not a quasigroup, 1-t=f is not invertible, but which is connected (1-t)X=X.

arXiv:1006.0258 [math.GT]

arXiv:1812.04704 [math.GT]

Date received: May 9, 2019


Some Problems in Topology
by
Paul C. Kainen
Georgetown University (Dept. of Math. and Stat.)

We discuss crossing numbers, book thickness, spatially embedded graphs, connected-sum cycle bases, and the Colin de Verdiere invariant.

It is shown that for a spatially embedded graph, if every member of a connected-sum cycle basis is flat, then the embedded graph is flat.

We conjecture that there exists a non-flat embedded graph with a flat (non-connected-sum) basis.

Date received: May 10, 2019


The Gram determinant of type Mb
by
Dionne Kunkel
The George Washington University
Coauthors: Rhea Palak Bakshi, Sujoy Mukherjee and Józef H. Przytycki

In 1995, a general closed formula for the Gram determinant of Type A was discovered in order to prove the existence and uniqueness of Lickorish's construct of the Witten - Reshetikhin - Turaev invariants of 3 - manifolds. This determinant is of a matrix given by a bilinear form on crossingless connections in the disc with 2n boundary points. Thirtheen years later, a general closed formula for the Gram determinant of Type B was solved. In this case, the determinant is of a matrix given by a bilinear form on crossingless connections in the annulus with 2n boundary points. The idea to work in the Möbius band, was formulated in October 2008. By April 2009, Qi Chen conjectured a general closed formula for the Gram determinant of the Möbius band, that is,


D(Mb)n(d, x, y, z, w) = Πi=0n Dn, iΠj=1nOn, j( 2n || (n-j )).
Where we let Dn, 0 = ∏k=1n (Tk(d)2-z2)(2n || (n-k )), and for i > 0, we let  Dn, i = ∏k=1+in (T2k(d)-2)( 2n || (n-k )), On, 2i = T2i(w) - [(2(2-z))/(T2i(d) - z)], and On, 2i+1 = T2i+1(w)- [2xy/(T2i+1(d) +z)].

Where (a || b) means a choose b and the i represents the number of curves passing through the cross cap.

In this talk, we will discuss the bilinear form on crossingless connections in the Möbius band with 2n boundary points then give insight to our progress in proving Qi Chen's conjecture.

Date received: May 7, 2019


A Big Data Approach to Knot Theory: The Polynomial Knot Invariants as Manifolds.
by
Jesse S F Levitt
University of Southern California
Coauthors: Mustafa Hajij (CoC), Radmila Sazdanovic (NCSU)

We examine the dimensionality and internal structure of the aggregated data produced by the Alexander, Bar-Natan and van der Veen, and Jones polynomials using topological data analysis and dimensional reduction techniques. By examining several families of knots, including over 10 million distinct examples, we find that the Jones data is well described as a three dimensional manifold, the Bar-Natan - van der Veen data as a two dimensional manifold and the Alexander data as a collection of two dimensional manifolds. These distinct results suggest the separate polynomials have different limits on their ability to distinguish between different knots. The ability to consider knots in this way illuminates several interesting relationships that I hope to discuss at the conclusion of the talk.

Date received: April 22, 2019


Counting factorizations in complex reflection groups
by
Joel Brewster Lewis
GWU
Coauthors: Alejandro Morales

In this talk, I'll discuss ongoing work with Alejandro Morales generalizing a 30-year old result of Jackson on permutation enumeration: we consider the enumeration of arbitrary factorizations of a Coxeter element in a well generated finite complex reflection group, keeping track of the fixed space dimension of the factors. As in the case of the symmetric group, the factorization counts are ugly, so the goal is to choose a basis for the generating function in which the answer is nice. In the case of the infinite families of monomial matrices, we accomplish this via combinatorial arguments; a notion of transitivity of a factorization appears for the "type D" group G(m, m, n). I'll also describe some puzzling partial results in the exceptional cases, and a tentative connection with maps on surfaces.

Date received: May 9, 2019


My ongoing struggle with quantum entanglement
by
Samuel Lomonaco
University of Maryland Baltimore County (UMBC)
Coauthors: Louis Kauffman

The talk will begin with a look at the GHZ Paradox, and then move on to discuss invariants of quantum entanglement.

Date received: May 2, 2019


Unknotting theta-curves and DNA replication
by
Danielle O'Donnol
Marymount University
Coauthors: Dorothy Buck

Replication is when a single DNA molecule is reproduced to form two new identical DNA molecules. In the middle of replication a more complex structure is formed. When a circular piece of DNA is replicated the intermediate structure is that of a theta-graph. A theta-graph has two vertices and three edges between them. This talk will focus on unknotting numbers of theta-graph and understanding DNA replication.

Date received: April 30, 2019


A new look at some old knots
by
Ken Perko
325 Old Army Road, Scarsdale, New York

We shall revisit some non-alternating knots mentioned briefly by Tait and Little and discuss why we think their 3-colored covering spaces are nice examples for beginners. References: Trans. Roy. Soc. Edinburgh 28 (1877), Plate XVI; idem 32 (1885), Plate LXXIX; idem 39 (1900), Plate III. See also Math. Ann. 24 (1884), Tafel XI.

Date received: April 29, 2019


The Affine Index Polynomial for virtual tangles. Preliminary report.
by
Nicolas Petit
Oxford College of Emory University

This is a preliminary report on current, work-in-progress research.

After reminding the audience of the affine index polynomial for virtual knots and its recent generalization to compatible links, both defined by Kauffman, we will discuss a version of the invariant for virtual tangles.

We will talk about the motivation for this invariant, go over its definition and why it generalizes the other affine index polynomials, and discuss a few properties/issues we're currently working on (mostly having to do with normalization).

Date received: April 22, 2019


An equivalence between gl(2)-foams and Bar-Natan cobordisms
by
Krzysztof Putyra
University of Zurich
Coauthors: Anna Beliakova, Matthew Hogancamp and Stephan Wehrli

The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way to associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms between planar trivalent graphs. Originating from the representation theory of quantum groups, foams are usually thought as algebraic objects. In my talk I will bring topology back by interpreting foams as two surfaces transverse to each other. This description leads to a quick proof that gl(2)-foams can be evaluated, a construction of a natural basis of foams, and an explicit equivalence between the category of gl(2)-foams and cobordisms between flat tangles [1]. An immediate application is a functorial version of the Chen-Khovanov tangle homology as well as of the quantized annular homology, constructed previously by Anna Beliakova, Stephan Wehrli, and me [2].

References: [1] arXiv:1903.12194 [2] arXiv:1605.03523

Date received: April 21, 2019


Invariants of annular links from categorical traces
by
Krzysztof Putyra
University of Zurich
Coauthors: Anna Beliakova, Stephan Wehrli

A trace function on a k-algebra A is a linear map tr: A→ k that satisfies the cyclicity relation tr(ab) = tr(ba). In particular, a trace of an A-valued invariant of tangles is automatically an invariant of links in a thickened annulus (which we call annular links). In a similar fashion one can obtain invariants of annular links from tangle homology theories, by replacing trace functions with their categorical analogues, such as Hochschild homology. Having such a description of an annular link homology one can then deform the trace relation to get a new, usually stronger, invariant of links in a solid torus.

Date received: May 10, 2019


Colored quantum gl(2) homology of links
by
Krzysztof Putyra
University of Zurich
Coauthors: Anna Beliakova, Stephan Wehrli, Matthew Hogancamp

Colored gl(2) homology is a categorification of the colored Jones polynomial of a link, a quantum sl(2) polynomial associated with a framed link decorated with a symmetric representation of sl(2). There are many non-equivalent ways how to construct such homology. For instance, Khovanov obtained a finite complex by considering sl(2) homology of cablings of a given knot and maps induced by annuli that contracts cables in pairs. The other approach due to Cooper and Krushkal starts with an infinite complex that categories the Jones-Wenzl projector. In my talk I will show that both constructions coincide if the quantum sl(2) homology is considered instead the usual one.

Date received: May 10, 2019


Unified Khovanov homology, their properties and computations
by
Alexander Shumakovitch
The George Washington University
Coauthors: Krzysztof Putyra

We give an overview of the unified Khovanov homology construction, focusing on their properties and computations.

Date received: May 10, 2019


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