Stable homotopy refinement of quantum annular homology
by
Rostislav Akhmechet
University of Virginia
Authors: Rostislav Akhmechet, Slava Krushkal, Michael Willis

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra, and Wehrli. Using an equivariant version of the Burnside category approach of Lawson-Lipshitz-Sarkar, we associate to an annular link $L$ a naive equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$. This is joint work with Slava Krushkal and Michael Willis.

Skein Modules Through the Ages
by
Rhea Palak Bakshi
The George Washington University
Authors: Rhea Palak Bakshi

This talk will give the audience a tour of the history and development of skein modules of 3-manifolds and their evolution into one of the most important tools in knot theory having connections to hyperbolic geometry, the SL(2,C) character variety and the Witten-Reshetikhin-Turaev topological quantum field theory. We will end the presentation with a discussion of Witten’s conjecture for Kauffman bracket skein modules.

Eulerian theorems for k-complexes
by
Paul C. Kainen
Georgetown University
Authors: Paul C. Kainen

We generalize to k-complexes Euler's basic theorem for connected graphs (even degree is equivalent to decomposability into edge-disjoint cycles is equivalent to the existence of a covering walk with no repeated edge). This is joint work with Richard Hammack.

Kashaev invariant and Kauffman states
by
Uwe Kaiser
Boise State University
Authors: Uwe Kaiser, Rama Mishra

For a given string knot universe we define and study a natural map from the set of Kashaev states to the set of Kauffman states (for the Alexander polynomial) with fixed *-regions and two fixed markers. This shows that the inequalities for colors resulting from Kashaev's vertex condition is completely encoded in Kauffman states. For the case of string braid closures we study the fibres of this map and discuss ordering the set of Kashaev states. In the case of weave knots explicit formulas can be given. We also discuss resulting questions concerning Kashaev state sums.

Foam evaluation and its applications
by
Mikhail Khovanov
Columbia University
Authors: Mikhail Khovanov

We'll review foam evaluation of Robert-Wagner and discuss its variations and applications, including to the categorification of HOMFLYPT polynomial and its extension to other level one representations and to Kronheimer-Mrowka homology theory for planar trivalent graphs.

SL(3) foams and Kronheimer-Mrowka theory
by
Mikhail Khovanov
Columbia University
Authors: M.Khovanov, L.-H.Robert

We go into more details on the unoriented SL(3) foam evaluation and connection to Kronheimer-Mrowka homology. The talk is based on a joint work with Louis-Hadrien Robert.

Foam evaluation and its applications II
by
Mikhail Khovanov
Columbia University
Authors: Mikhail Khovanov

We'll continue discussing foam evaluation and its applications to link homology theories.

Spaces of knots in the solid torus and irreducible links in the 3-sphere
by
Robin Koytcheff
University of Louisiana at Lafayette
Authors: Andrew Havens and Robin Koytcheff

Motivated by a question of Arnold, we recursively determine the homotopy type of the space of any knot in the solid torus. We proceed by establishing results on spaces of irreducible links in the 3-sphere, modulo rotations. These results generalize work of Hatcher and Budney. We also obtain descriptions of spaces of knots in the thickened torus. In all three cases, we find splittings of certain subgroups of rotations at the level of the fundamental group. Their generators can be viewed as variants of the Gramain loop in the space of long knots.

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

In the 1990's, a general formula for the Gram determinant of Type A was formulated 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 crossless connections in the disc with $2n$ boundary points. A decade later, general formula for the Gram determinant of Type B was solved. We generalize the Gram determinant of type A and formulate a closed formula as well as prove results that support Qi Chen's conjecture of a general closed formula for the Gram determinant of the M\""obius band.

Reidemeister moves as simple homotopy moves
by
Samuel Lomonaco
University of Maryland, Baltimore County
Authors: Samuel Lomonaco

We explain how to consider Reidemeister moves as simple homotopy moves in higher dimensions

Studying Links with Mark E. Kidwell
by
Kerry M Luse
Trinity Washington University
Authors: Kerry M Luse

In this talk, I will describe my experiences working with Mark and highlight some of our work together. I will also discuss our most recent work which showed certain terms of the Alexander polynomial $\Delta(x,y)$ of a rational link are related to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. If the rational link has a reduced alternating diagram with no self-crossings, then $\Delta(-1,0)=1$. If the standard form of the rational link has $M$ monochromatic twist sites, and the $j$th monochromatic twist site has $m_j$ crossings, then $\Delta(-1,0)= \prod_{j=1}^M(m_j+1)$. Finally, I will discuss our conjecture that $\Delta(-1,0)=1$ holds for all 2-component alternating links with no self-crossings.

Skein Modules and Framing Changes of Links in $3$-Manifolds.
by
Gabriel Montoya-Vega
George Washington University
Authors: Rhea Palak Bakshi, Dionne Ibarra, Gabriel Montoya-Vega, Jozéf Przytycki, and Deborah Weeks.

The framing of a knot K in an oriented $3$-manifold remains unchanged unless there exists a properly embedded non-separating $2$-sphere which intersects K exactly once; in which case, the change of framing is given by the Dirac trick. Since 1987, when Przytycki introduced skein modules, these have been extensively studied with the goal of building an algebraic topology based on knots. In this talk, we formulate the results on the framing of knots in the language of skein modules.

Homotopy type of disk complexes associated to bridge spheres for links
by
Puttipong Pongtanapaisan
University of Iowa
Authors: Puttipong Pongtanapaisan and Daniel Rodman

Consider a link in the 3-sphere in bridge position. Compressing disks for the bridge sphere give rise to a simplicial complex called the disk complex. David Bachman showed that in an irreducible 3-manifold, a surface with non-contractible disk complex and an incompressible surface can be isotoped so that any intersection loop is essential in both surfaces. In this talk, I will demonstrate a retraction from the disk complex of a bridge sphere for a nontrivial link to a high-dimensional sphere. This is joint work with Daniel Rodman.

The second homology of Homflypt Yang-Baxter operators
by
Jozef H. Przytycki
GWU
Authors: Jozef H. Przytycki, Xiao Wang

We discuss two contradictory conjectures on the structure of the Yang-Baxter homology of the column unital Yang-Baxter operators yielding the Homflypt polynomial. The main result is the closed formula for the second homology. We have also many numerical calculations and a conjecture for the concrete closed formula for the level 2 (that is giving the Jones polynomial) Yang-Baxter homology.

In Praise of Diagrams
by
Heather M. Russell
University of Richmond
Authors: Heather M. Russell

Diagrams play an important role in mathematics. Not only do they provide a convenient framework for computation, but also they may lead to new insights into the object of study and surprising connections between seemingly disparate areas of math. Perhaps nowhere is the role of the diagram more elevated than in knot theory. In this talk, we will discuss the importance of diagrams with a focus on three examples: knot diagrams, webs, and Young tableaux.

Highest-weight representations and global Weyl modules: from classical Lie algebras to Yangians
by
Prasad Senesi
The Catholic University of America
Authors: Prasad Senesi*, Bach Nguyen and Matt Lee

Highest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Particular examples of highest–weight representations of certain infinite–dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding Loop algebras and quantum groups. Along the way, the utility of highest–weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago).

On the complexity of computing Khovanov homology
by
Marithania Silvero
Universidad de Huelva
Authors: Józef Przytycki

Khovanov homology is a link invariant which categorifies Jones polynomial. In general, computing Jones polynomial (so also Khovanov homology) is NP-hard.

However, if we consider a closed braid of fixed number of strands, it is well-known that all classical quantum invariants (in particular Jones polynomial) can be computed in polynomial time.

We conjecture that the complexity of computing Khovanov homology of a closed braid of fixed number of strands, is polynomial with respect to the number of crossings. In this talk we show some advances on the conjecture, showing that the result holds when considering extreme Khovanov homology of closed braids on 3 and 4 strands. This is an ongoing project.

Algorithmic complexity of properties of structures
by
Dario Verta
GWU
Authors: Valentina Harizanov, Darío Verta

In computable model theory, we are often interested in the algorithmic complexity associated with finding algebraic structures that exhibit interesting properties. Given the ubiquity of groups in modern mathematics, they are a natural candidate of study. Novikov and Boone's (independent) results on the undecidability of the word problem has motivated related questions pertaining to the difficulty of identifying so-called Markov properties in certain classes of groups. In similar form to the work of Rabin and Lockheart, we'd like to present some old and new results that allow us to further describe the algorithmic complexity for classes of recursively presented groups and, more generally, for magmas.

Idempotent solutions of the Yang-Baxter equation
by
Petr Vojtechovsky
University of Denver
Authors: David Stanovsky and Petr Vojtechovsky*

We will describe all latin idempotent solutions of the Yang-Baxter equation up to isomorphism and prove that they can be thought of as esentially the left division in groups with a twist by an automorphism. In the more general left non-degenerate case of idempotent solutions, we describe the corresponding algebraic objects and provide complete classification for prime orders. This is an expanded version of my talk at the Denver Joint Mathematics Meeting.

Framing Changes of Links in 3-Manifolds
by
Deborah J weeks
George Washington University
Authors: Rhea Palak Bakshi, Dionne Ibarra, Gabriel Montoya-Vega, Jozéf Przytycki, and Deborah Weeks.

Extending the work of McCullough and Chernov, the conditions under which the framing of a link can be changed via ambient isotopy are examined. A change in framing can only be accomplished via the Dirac trick in a manifold with a properly embedded non-separating $S^2$. Due to the invariance of spin structure and the parallelizability of every compact oriented 3-manifold the framing of a knot can only change by an even power.