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Organizers |
"Knots, Links and Geometry"
by
Herman Gluck
University of Pennsylvania
Coauthors: Dennis DeTurck, Rafal Komendarczyk, Paul Melvin, Haggai Nuchi, Liu-Hua Pan,
Clayton Schonkwiler, David Shea Vela-Vick, Peter Storm
The trajectory of my mathematical life began with knots and low-dimensional topology, then moved into differential geometry, and in recent years is once again involved with knots and links, but this time from the perspective of geometry, and with application to molecular biology, topological fluid dynamics and plasma physics.
I will survey this most recent chapter of my work.
Some key words and phrases: knot theory in the presence of curvature, Gauss linking integral, writhing of knots, helicity of vector fields, upper bounds for the writhing of DNA, lower bounds for the energy of magnetic fields, the search for higher order helicities, link homotopy, generalized Gauss maps for three-component links, Lipschitz minimality of Hopf maps and vector fields.
Date received: April 11, 2013
Heegaard Floer homology and Strip Diagrams
by
Carl Hammarsten
George Washington University
A 3-dimensional closed manifold Y represented by its branched spine has a canonical Heegaard decomposition. We present this decomposition graphically in the form of a Strip Diagram. We show that strip diagrams have nice properties which greatly simplify the calculation of Heegaard Floer homology for certain manifolds. We hope that this construction works for all closed 3-manifolds.
Date received: April 30, 2013
A special case of the WRT invariant and Khovanov homology
by
Noboru Ito
Waseda Institute for Advanced Study
Setting that the variable of the Kauffman bracket is the 20th root of unity (i.e. the case that r = 5), we show the invariance of a direct sum of Khovanov homology groups under handle slides. This direct sum is induced by the Jones-Wenzl projector. This Khovanov homology is given as a categorification of a liner sum of colored Jones polynomials corresponding to the WRT invariant. We also discuss about a behavior of the Khovanov homology under the first Kirby moves.
Date received: April 13, 2013
Khovanov Homology for Virtual Knots and Links
by
Louis H. Kauffman
University of Illinois at Chicago
Coauthors: Heather Dye and Aaron Kaestner
We show how to define integral Khovanov homology for virtual knots and links by using
a source-sink orientation on the projected link diagram. This gives an alternate apporach to the
method of Vassily Manturov.
Date received: April 9, 2013
Catalan States of Generalized Crossings
by
Changsong Li
Department of Mathematical Science, University of Texas at Dallas
Coauthors: Mieczyslaw Dabkowski (UTD) &
Jozef Przytycki (GWU)
The main motivation for this work is a problem of finding a closed formula for the product in the Kauffman Bracket Skein Algebra of I-boundle over 3 punctured disk. However, the presentation for this algebra was found by D. Bullock and J. Przytycki and special cases of the formula for product are maybe not hard to find, but finding an explicit general formula for product is still an open problem which leads to many interesting questions. One of them is the problem of finding Kauffman Bracket for a (n+m)-tangle D(m, n) with m horizontal parallel arcs placed over n parallel vertical parallel arcs. In this talk, we will present results of our joint work with M. Dabkowski and J. Przytycki which show that the horizontally excluded Kauffman states for D(m, n) have a description in terms of lattice paths and their number Th(m, n) can be found.
Date received: April 28, 2013
Seifert forms, the Alexander module, and bordered Floer homology
by
Tye Lidman
UT Austin
Coauthors: Jennifer Hom, Sam Lewallen, Liam Watson
Knot Floer homology is a useful invariant of knots whose graded Euler characteristic recovers the Alexander polynomial. Some other objects which can recover the Alexander polynomial are the Seifert form and the Alexander module. We will discuss how the bordered Floer homology of the complement of a Seifert surface for a knot in the three-sphere can be seen as a common refinement of all of these invariants. No familiarity with Floer homology will be assumed.
Date received: April 21, 2013
The Geometry of Fox's Free Calculus and Higher Dimensional Knots
by
Samuel J. Lomonaco
University of Maryland Baltimore County & Princeton University
We discuss the geometry hidden within Fox's Free Calculus, and Its Application to Higher Dimensional Knots.
Date received: April 21, 2013
Progress in one term distributive homology
by
Jozef H. Przytycki
George Washington University
Coauthors: Alissa S.Crans, Krzysztof K.Putyra
I will discuss the progress made in the study of one term distributive homology made in the last few months. We computed completely the one term distributive homology for a spindle of 2 block decomposition of the type X=X0\sqcup b. An important tool is the simplified version of the Künneth formula for the degenerate part of rack homology of spindles (see K.Putyra talk). We also show that one term distributive homology of a finite spindle can have any finite torsion. The first example in which a torsion contains Z2⊕Z4, the motivation for the general, multi-block construction, is given below:
Consider a 8-element spindle (X;*) with operation * given by the following table
(notice blocks of size 5 and 3):
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Similarly consider a 17-element spindle (X;*) with operation * given by the following table (notice blocks of size 3, 5 and 9), to get H1(X) = Z20 ⊕Z2 ⊕Z4 ⊕Z8.
Paper reference: http://front.math.ucdavis.edu/1109.4850, http://arxiv.org/abs/1111.4772, http://front.math.ucdavis.edu/1105.3700
Date received: May 1, 2013
Degenerate distributive homology is degenerate indeed
by
Krzysztof K. Putyra
Columbia University
Coauthors: Jozef H. Przytycki
The distributive homology of quandles and racks proved to be a useful tool in the theory of knots and links. After its discovery by Fenn, Rourke and Sanderson and then by Carter, Kamada and Saito, it was noticed that the distributive chain complex for a quandle splits into two parts, normalized and degenerate, immitating the simplicial homology theory. However, a degenerate complex is not acyclic, contrary to the simplicial theory. Recently, with Jozef Przytycki we managed to prove that the degenerate part, as its name suggests, is really degenerate: it is completely determined by the normalized homology.
In my talk I will describe the distributive chain complex associated to a quandle and, more generally, to a spindle, and how its splitting into degenerate and normalized parts. Then I will give a recursive formula for the degenerate part and sketch main ideas of its proof.
Date received: April 19, 2013
Towards unifying group and quandle homology theories
by
Masahico Saito
University of South Florida
Coauthors: J. Scott Carter, Atsushi Ishii
Quandle homology theories have been constructed in analogy of group homology. There are quandles in which associative operations are partially defined, and the two operations satisfy certain compatibility conditions. We present such structures, and propose a homology theory that reflect both operations. Their colorings and cocycles are used to define invariants for handle-body links.
Date received: April 21, 2013
Bendings by finitely additive transverse cocycles
by
Dragomir Saric
CUNY/Graduate Center and Queens College
Let S be a closed hyperbolic surface and let L be a maximal geodesic lamination on S. Thurston and Bonahon parametrized pleated surfaces with the pleating locus L using finitely additive complex-valued transverse cocycles to L. A pleated surface with the pleating locus L is obtained by bending a (totally geodesic in the hyperbolic three space) hyperbolic structure according to a transverse cocycle. Motivated by the recent proof of the surface subgroup conjecture(Kahn-Markovic), we establish a sufficient condition on transverse cocycles such that the bending map induces a quasiFuchsian representation of the fundamental group of S. Our condition is genus independent.
Date received: May 2, 2013
Homogeneous links and two related families
by
Marithania Silvero-Casanova
Departamento de Álgebra - Universidad de Sevilla (España)
Peter Cromwell introduced homogeneous links, which include positive and alternating links. They also include alternative links, introduced by Kauffman. Pseudo-alternating links, in the sense of Mayland and Murasugi, contain homogeneous links. Kauffman conjectured that pseudo-alternating links are alternative, implying that the three families (alternative, homogeneous and pseudo-alternating links) are equivalent.
I will define these three families of links, showing some inclusions involving them and giving some examples.
Date received: April 26, 2013
Realizing Khovanov Homology as Homology of a Small Category with Functor Coefficients
by
Jing Wang
George Washington University
Coauthors: Jozef H. Przytycki
In this talk, I will introduce our joint work with Jozef H. Przytycki about how to realize Khovanov homology in the context of homology of a small category with functor coefficients. Our general definition is motivated by homology of an abstract simplicial complex. In particular, I will show that for an abstract simplicial complex category, the theory we introduced here can be considered as homology of its barycentric subdivision. I will give a short proof showing that in the twisted coefficient case, homology is preserved under barycentric subdivision as in the classical theory.
Date received: April 29, 2013
Equivariant Khovanov-Rozansky Homology and Lee-Gornik Spectral Sequence
by
Hao Wu
George Washington University
In this talk, I will:
1. explain why an equivariant sl(N) Khovanov-Rozansky homology over C[a] is equivalent to the corresponding Lee-Gornik spectral sequence,
2. give an alternative construction of the Lee-Gornik spectral sequence using a simple exact couple,
3. define a natural exterior algebra action on the sl(N) Khovanov-Rozansky homology,
4. make a few other observations about the Khovanov/Khovanov-Rozansky homology.
All the above are based on a simple observation by Lobb on the module structure of an sl(N) equivariant Khovanov-Rozansky homology over C[a].
Date received: April 29, 2013
Abelian quotients of the string link monoid
by
Akira Yasuhara
Tokyo Gakugei University
Coauthors: Jean-Baptiste Meilhan (University of Grenoble 1)
The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n=1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the Ck-equivalence introduced by Habiro in connection with finite type invariants theory, and the Ck-concordance, which is generated by Ck-equivalence and concordance. We investigate under which condition these groups are abelian.
Date received: April 28, 2013
Copyright © 2013 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.