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Colored Khovanov-Rozansky homology of infinite braids
by
Michael Abel
Duke University
Coauthors: Michael Willis (University of Virginia)
Islambouli and Willis showed in their recent work that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. In this talk we show that the limiting Khovanov-Rozansky sl(n) chain complex of any infinite positive braid categorifies the highest weight projectors, generalizing an earlier result of Cautis. We also show that an analogous result holds in the unicolored case. That is, when one colors all of the strands by the same fundamental representation of sl(n).
Date received: April 14, 2017
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: April 19, 2017
Irregular dihedral covers of the four-sphere and linking numbers in 3-manifolds
by
Patricia Cahn
Smith College
Coauthors: Alexandra Kjuchukova
We give an algorithm for computing the linking number of two zero-homologous knots in a closed oriented 3-manifold, where the 3-manifold is presented as an irregular 3-fold dihedral cover of S^3 branched along a knot. Our method generalizes an algorithm of Perko. We then describe how these linking numbers, together with a formula of Kjuchukova, can be used to compute signatures of 4-manifolds which are 3-fold irregular dihedral covers of S^4.
Date received: April 25, 2017
A Categorification of the Vandermonde Determinant
by
Alex Chandler
NCSU
In the spirit of Bar Natan's construction of Khovanov homology we introduce a diagrammatic framework for categorifying the Vandermonde determinant. We form a diagram of colored smoothings and cobordisms of a knot diagram in the shape of the Hasse diagram of the Bruhat order on the symmetric group. Applying a TQFT to this diagram yields a chain complex whose Euler characteristic is equal to the Vandermonde determinant.
Date received: April 18, 2017
On the extended bracket polynomial for virtual knots and links
by
Sergei Chmutov
Ohio State University, Mansfield
I will report a result of my student Benjamin O'Connor obtained in a summer program "Knots and Graphs" in 2012:
https://people.math.osu.edu/chmutov.1/wor-gr-su12/wor-gr.htm
He found a gap in L.Kauffnam's proof of isotopy invariance of the extended bracket polynomial for virtual
knots and links. I will show his counterexample and explain why the proof does not work.
Date received: April 11, 2017
Properties of Cohesive Powers
by
Rumen Dimitrov
Department of Mathematics, Western Illinois University, Macomb IL 61455
Coauthors: Some of the work is based on a manuscript with co-authors R. Dimitrov, P. Shafer, A. Soskova, and S. Vatev.
Cohesive powers of computable fields were used (in [1]) to characterize the principal filters of quasimaximal spaces with extendable bases in the lattice L*(V∞) . In [2] we used the isomorphism properties of cohesive powers of Q to classify the orbits of the equivalence classes of such spaces in the lattice L*(V∞). In this talk I will prove different model theoretic properties of the cohesive powers of various computable structures.
[1] R.D. Dimitrov, A class of Σ30 modular lattices embeddable as principal filters in L*(V∞), Archive for Mathematical Logic 47 (2008), pp. 111-132.
[2] R.D. Dimitrov and V. Harizanov, Orbits of maximal vector spaces, Algebra and Logic 54 (2015), pp. 680-732 (Russian); (2016) pp. 440-477 (English translation).
[3] R.D. Dimitrov, P. Shafer, A. Soskova, and S. Vatev, Notes on cohesive powers and Fraïssé limits, in preparation.
Date received: April 2, 2017
Distinguishing Legendrian knots having trivial topological symmetry group
by
Ivan Dynnikov
Steklov Mathematical Institute of Russian Academy of Sciences
Coauthors: Vladimir Shastin (Moscow State University)
Each rectangular diagram of a knot (aka grid diagram or arc-presentation) naturally defines two Legendrian knots, one with respect to the standard contact structure and the other with respect to its mirror image. If K is a topological type of a knot that has no non-trivial symmetry we show that two rectangular diagrams R1 and R2 representing K are related by a finite sequence of exchange moves if and only if the Legendrian knots defined by R1 are equivalent to the respective Legendrian knots defined by R2.
This allows in certain cases to relatively easily distinguish Legendrian knot types that cannot be distinguished by any known algebraic invariants either because the invariants are equal for the knots or because they are too hard to compute. In particular, we prove the existence of an annulus A tangent to the contact structure along the whole boundary, such that the two boundary components of A are not equivalent as Legendrian knots. We use the concrete example of such an annulus that was suggested previously by I.Dynnikov and M.Prasolov.
The work is based on a recent study of rectangular diagrams of surfaces by I.Dynnikov and M.Prasolov.
Date received: April 10, 2017
Minimal Intersection Number of Flat Virtual Links
by
David Freund
Dartmouth College
Given two free homotopy classes of loops on a surface, it is natural to count the minimal number of intersection points up to stabilization and destabilization of the surface. We use generalizations of the Goldman Lie bracket and the AndersenMattesReshetikhin Poisson bracket to obtain lower bounds on this quantity.
Date received: April 10, 2017
Not every graph has a robust cycle basis.
by
Richard H. Hammack
Virginia Commonwealth University
Robust cycle bases of graphs have both theoretical and biological applications. However, robust cycle bases are known to exist only for a few classes of graphs. Despite this, previously no graph was known to not have a robust cycle basis. Theorem. The complete bipartite graphs Kn, n have no robust cycle basis when n ≥ 8. Open questions, including the gap 5 ≤ n ≤ 7, and for other graphs are described. This is joint work with Paul Kainen.
Date received: April 19, 2017
Topology and commutativity of diagrams
by
Paul C. Kainen
Georgetown University
Robustness and iterative robustness of cycle bases are defined. The proof that some graphs cannot have a robust basis is given in the next talk by Richard Hammack, with whom this work is joint. We show that diagrams commute iff they are commutative when restricted to an iteratively robust basis of the underlying graph of the diagram. Theorem. Every graph has an iteratively robust basis. A non-commutative diagram is exhibited whose underlying graph has a cycle basis with all members commutative. So iterative robustness of the basis matters for the control of commutativity.
Date received: April 19, 2017
Revisiting an Extended Bracket Polynomial for Virtual Knots
by
Louis H Kauffman
University of Illinois at Chicago
We consider a bracket expansion where one replaces an oriented crossing with combination of a smoothed crossing (s) and a disoriented smoothed crossing (dsc). The dsc has the appearance of two cusps meeting one another. One cusp has arrows pointing toward one another. One cusp has arrows pointing away from each other. The two cusps are said to be paired if they are in the form of the dcc. Paired cusps form a species of decorated 4-valent vertex. We give a set of graphical replacements for certain configurations of paired cusps so that the bracket state expansion based on < K+ > = A < Ks > + A-1 < Kdsc > is an invariant of regular isotopy where the graphs that appear in the expansion are taken up to the equivalence relation generated by the graphical replacements. A special case of this invariant is the arrow polynomial, where the cusps do not have to be paired. We discuss the differences between this extended invariant and the arrow polynomial. This discussion forms the basis for further technical discussions about errors (discovered by Sergei Chmutov and Bejamin OConnor) in a previous paper by the author where he erroneously postulated a specific ordered reduction procedure for these graphs based on the above mentioned graphical moves. One can formulate a basic invariant using an equivalence relation rather than a reduction procedure and then start the investigation again in this new way.
Date received: April 26, 2017
Delta-matroids and Vassiliev invariants
by
Sergei Lando
Higher School of Economics, Skolkovo Institute of Science and Technology
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. There is also a natural way to define 4-term relations for abstract graphs, and graph invariants satisfying these relations produce weight systems: to each chord diagram its intersection graph is associated.
The notion of weight system can be extended from chord diagrams, which are orientable embedded graphs with a single vertex, to embedded graphs with arbitrary number of vertices that can well be nonorientable. These embedded graphs are a tool to describe finite order invariants of links: the vertices of a graph are in one-to-one correspondence with the link components.
We are going to describe two approaches to constructing analogues of intersection graphs for embedded graphs with arbitrary number of vertices. One approach, due to V. Kleptsyn and E. Smirnov, assigns to an embedded graph a Lagrangian subspace in the relative first homology of a 2-dimensional surface associated to this graph. Another approach, due to S. Lando and V. Zhukov, replaces the embedded graph with the corresponding delta-matroid, as suggested by A. Bouchet in 1980's. In both cases, 4-term relations are written out, and Hopf algebras are constructed.
Vyacheslav Zhukov proved recently that the two approaches coincide.
Date received: April 8, 2017
Reducing 4-D Knot Theory to 3-D knot Theory
by
Samuel J. Lomonaco
University of Maryland Baltimore County (UMBC)
We show how to reduce the study of knotted surfaces in 4-space to a study of knots in 3-space.
Date received: April 18, 2017
The Alexander Polynomial of a Rational Link
by
Kerry M. Luse
Trinity Washington University
Coauthors: Mark E. Kidwell
We relate some terms on the boundary of the Newton polygon of the Alexander polynomial Δ(x, y) of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize Δ(x, y) so that no x-1 or y-1 terms appear, but x-1Δ(x, y) and y-1Δ(x, y) have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then Δ(-1, 0) = 1. If the standard form of the rational link has m monochromatic twist sites, and the jth monochromatic twist site has q̂j crossings, then Δ(-1, 0) = ∏j=1m(q̂j+1). Our proof employs Kauffman's clock moves and a lattice for the terms of Δ(x, y) in which the y-power cannot decrease.
Date received: April 6, 2017
Representation theory for racks
by
Elkaioum Moutuou
University of South Florida
Coauthors: Mohamed Elhamdadi
This is an introduction to representation theory or racks and quandles.
I will focus on the specific class of "finitely stable racks" and
show how much they share with Abelian groups.
The talk will be based on our article avalaible on the ArXiv: https://arxiv.org/pdf/1611.04453v1.pdf
Date received: April 18, 2017
Lbo homology and Jones monoids
by
Sujoy Mukherjee
The George Washington University
I will start with a brief introduction to lbo homology using pre-simplicial modules and then discuss some of its properties. Finally, I will describe the connection between lbo homology and Jones monoids and my attempts to connect it with knot theory.
Date received: April 26, 2017
A Product Structure on Generating Family Cohomology for Legendrian Submanifolds
by
Ziva Myer
Bryn Mawr College, Duke University
In contact geometry, invariants of Legendrian knots in R3, and more generally, Legendrian submanifolds in 1-jet spaces, have been obtained through a variety of techniques. I will discuss how I am extending one such invariant, Generating Family Cohomology, by constructing a product structure. The construction uses moduli spaces of Morse flow trees spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. This product lays the foundation for an A-infinity algebra that will show, in particular, that Generating Family Cohomology has an invariant ring structure.
Date received: April 10, 2017
Topological Mythology
by
Kenneth A. Perko, Jr.
Uninstitutionalized
We shall discuss some published misconceptions about the Tait conjectures, Reidemeister moves and Fox coloring. In short, Tait never made them, Maxwell introduced them, and Reidemeister discovered it. Cf. page 7 of J. H. Przytycki's "The Trieste look at Knot Theory" [arXiv:1105.2238v1 [math GT] 11 May 2011].
Date received: March 10, 2017
Mirror diagrams and Legendrian equivalence
by
Maxim Prasolov
Moscow state university
Coauthors: Ivan Dynnikov
Mirror diagrams are a special way to represent spatial ribbon graphs. Together with elementary moves they provide a nice calculus for describing isotopy classes of surfaces in the three-space. The elementary moves are of two types and the main fact about them is that moves of different types commute in some sense. A new method for distinguishing Ledenrian knots is based on this fact.
Date received: April 14, 2017
Almost extreme Khovanov homology of semi-adequate diagrams
by
Jozef H. Przytycki
George Washington University
Coauthors: Marithania Silvero (IMPA)
We say that (Xn, di) is a partial presimplicial set if
(1) (kXn, di) is a presimplicial module, that is didj=dj-1di for i < j, and
(2) for any xn ∈ Xn we have di(xn) ∈ Xn-1 or di(xn)=0.
Almost presimplicial set allows a standard geometric realization: if di(xn)=0 then the ith face of xn ×Δn is contracted. We show that almost extreme Khovanov homology of a A-adequate link can be obtained from an almost presimplicial set giving a finite CW complex geometric realization. In particular we show that for the trefoil knot its geometric realization is a projective plane, RP2. We conjecture that the geometric realization will be homotopy equivalent either to Sm or the suspension Σm-2RP2 depending on whether the B- state graph is bipartite or contains an odd cycle. m+1 is the number of crossings of considered link diagram. For example for the figure eight knot we obtain ΣRP2. We outline a proof of the conjecture which is based on the previous work of R.Sazdanovic and M.Silvero with the author.
Paper reference: arXiv:1608.03002 [math.GT], arXiv:1210.5254 [math.QA]
Date received: April 26, 2017
Torsion in Khovanov link homology via chromatic graph cohomology
by
Dan Scofield
North Carolina State University
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 Z_2 torsion of certain classes of knots and links.
Date received: April 10, 2017
Classification of the Endofunctors of the Algebraic Theories of Quandles and Racks
by
Peter Ulrickson
The Catholic University of America
We describe the endofunctors of the categories of racks and quandles which commute with the forgetful functor to sets.
Date received: April 27, 2017
Khovanov homology and knots and links with smoothing number one.
by
Xiao Wang
The George Washington University
Coauthors: Sujoy Mukherjee, Józef H. Przytycki, Marithania Silvero and Seung Yeop Yang
In the Khovanov homology of links, appearance of torsion other than Z2 is rare. Based on the examples of torus links with torsion other than Z2, we found infinite families of links with Zn-torsion for 2 < n < 9 and Z2s-torsion for s < 24
Date received: April 27, 2017
Knot invariants and Hecke correspondences
by
Ben Webster
University of Virginia
The Jones polynomial and other Reshetikhin-Turaev invariants are blessed (or depending on your perspective, cursed) with two complementary and quite different descriptions: Witten has written them as expectation values for Wilson loop operators in Chern-Simons theory, but in order to effectively compute them, we can exploit dimensional reduction which gives us a description in terms of very simple tangles (using quantum groups).
Much more recently, Witten has proposed a similar approach to understanding Khovanov homology using field theories (essentially the counting of solutions to PDEs). I'll try to give a mathematician's perspective on this theory, and the dimensional reduction approach to computing it. Just as quantum groups spring fully formed from the forehead of Chern-Simons theory, this approach gives us another perspective to understand the constructions of homological knot invariants, including KLR algebras.
Date received: April 17, 2017
Torsion in Khovanov homology of twist deformations of torus links
by
Seung Yeop Yang
George Washington University
Coauthors: Sujoy Mukherjee, Józef H. Przytycki, Marithania Silvero and Xiao Wang
We analyze Khovanov homology of twist deformations of torus links and show that counterexamples to the PS braid conjecture can be obtained in this procedure. Moreover, we provide some examples showing that the Khovanov homology of the flat 2-cabling of a given knot may contain interesting torsion subgroups.
Date received: April 27, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.