|
Organizers |
Categorified Young antisymmetrizers and stable HOMFLYPT homology of torus links
by
Michael Abel
Virginia Commonwealth University
Coauthors: Matt Hogancamp
The HOMFLY-PT polynomial colored by one-column partitions can be constructed by cabling a braid by objects known as Young antisymmetrizers in the Hecke algebra of Sn. In this talk, we will explictly show how to categorify the Young antisymmetrizers in the homotopy category of Soergel bimodules in the case n = 2 and 3. Along the way we will discuss properties which can be generalized to any n, leading to a more general construction. We will also discuss the relation between the categorified Young antisymmetrizers and the stable HOMFLYPT homology of torus links.
Date received: February 25, 2015
The Minimal Zn-Symmetric Graphs That Are Not Zn-Planar
by
Lowell Abrams
The George Washington University
Coauthors: Daniel Slilaty (Wright State University)
Given a graph G equipped with fixed-point-free Γ-action (Γ a finite group) we define an orbit minor H of G to be a minor of G for which the deletion and contraction sets are closed under the Γ-action. The orbit minor H inherits a Γ-symmetry from G, and when the contraction set is acyclic the action inherited by H remains fixed-point free. When G embeds in the sphere and the Γ-action on G extends to a Γ-action on the entire sphere, we say that G is Γ-spherical. We determine for every odd value of n ≥ 3 the orbit-minor-minimal graphs G with a free Zn-action that are not Zn-spherical. There are 11 infinite families of such graphs, each of the 11 having exactly one member for each n. For n=3, another such graph is K3, 3. The remaining graphs are, essentially, the Cayley graphs for Zn aside from the cycle of length n. The result for n=1 is exactly Wagner's result from 1937 that the minor-minimal graphs that are not embeddable in the sphere are K5 and K3, 3.
Date received: March 2, 2015
Scaffold assembly based on genome rearrangement analysis
by
Sergey Aganezov
Computational Biology Institute & Department of Mathematics, George Washington University
Coauthors: Nadia Sitdykovaa, AGC Consortium, Max A. Alekseyev
Advances in DNA sequencing technology over the past decade have increased the volume of raw sequenced genomic data available for further assembly and analysis. While there exist many algorithms for assembly of sequenced genomic material, they often experience difficulties in constructing complete genomic sequences. Instead, they produce long genomic subsequences (scaffolds), which then become a subject to scaffold assembly aimed at reconstruction of their order along genome chromosomes. The balance between reliability and cost for scaffold assembly is not there just yet, which inspires one to seek for new approaches to address this problem.
We present a new method for scaffold assembly based on the analysis of gene orders and genome rearrangements in multiple related genomes (some or even all of which may be fragmented). Evaluation of the proposed method on artificially fragmented mammalian genomes demonstrates its high reliability. We also apply our method for incomplete anophelinae genomes, which expose high fragmentation, and further validate the assembly results with referenced-based scaffolding. While the two methods demonstrate consistent results, the proposed method is able to identify more assembly points than the reference-based scaffolding.
Date received: February 19, 2015
Breakpoint graph: transposition rate
by
Nikita Alexeev
Computational Biology Institute, George Washington University
Coauthors: Anna Pologova, Rustem Aidagulov, Max A. Alekseyev
Genome rearrangements are evolutionary events that shuffle genomic architectures. Most frequent genome rearrangements are reversals, translocations, fusions, and fissions. While there are some more complex genome rearrangements such as transpositions, they are rarely observed and believed to constitute only a small fraction of genome rearrangements happening in the course of evolution. We propose a computational method for estimating the rate of transpositions in evolutionary scenarios between two genomes. In this talk we describe genome rearrangements in terms of graphs, define a breakpoint graph, corresponding to a pair of genomes, and explain, how its cycle structure allows to estimate the transposition rate. If we have enough time, we will also discuss an application of topological recursion to the enumeration of breakpoint graphs with a given structure.
Date received: February 24, 2015
Braided molecules
by
Senja Barthel
Imperial College London, UK
Coauthors: Davide M. Proserpio, F. Din-Houn Lau, Igor Baburin
Coordination polymers (think of crystals) can occur in topologically different modes. The modes of 1-dimensional coordination polymers have a description as braids. We predict all possible entanglements of those molecules combinatorial and present a method to compare them with real chemical structures.
Date received: February 17, 2015
Knots transverse to a vector field
by
Patricia Cahn
University of Pennsylvania
Coauthors: Vladimir Chernov
We study knots transverse to a fixed vector field V on a 3-manifold M up to the corresponding isotopy relation. We show this classification is particularly simple when V is the co-orienting vector field of a tight contact structure, or when M is irreducible and atoroidal. We also apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges.
Date received: February 27, 2015
A formula for the Dubrovnik polynomial of rational knots
by
Carmen Caprau
California State University, Fresno
Coauthors: Katherine Urabe
The Dubrovnik polynomial is a version of the two-variable Kauffman polynomial for unoriented knots and links. In this talk, we derive a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.
This work was inspired by a paper of S. Duzhin and M. Shkolnikov, in which they gave an explicit formula for the HOMFLY-PT polynomial of oriented rational knots in terms of a continued fraction for the rational number that represents the given knot.
Date received: February 27, 2015
Remembering Sergei Duzhin
by
Sergei Chmutov
Ohio State University, Mansfield
I will recall life and work of my life long friend Sergei Duzhin who suddenly passed away on February 1, 2015 from sharp heart failure. The main topic of our collaboration with Sergei was Vassiliev knot invariants. So the mathematical part of my talk will concentrate on this theory. I will give first definitions, survey the theory, and explain our results as well as their developments. Very briefly I recall his life and other his results.
Date received: February 22, 2015
Annuli with Legendrian boundary
by
Ivan Dynnikov
Steklov Mathematical Institute, Moscow
Coauthors: Maxim Prasolov
Let A be an annulus embedded in the three-space so that A is tangent to the standard contact structure at all boundary points. This means, in particular, that A is cobounded by two knots K1, K2 that are Legendrian and have the same topological type. Is it true that K1 and K2 are always Legendrian equivalent? By using grid diagrams we prove a weaker statement and suggest a method to disprove the original one.
Date received: February 19, 2015
Heegaard Floer homology of some L-space links
by
Eugene Gorsky
Columbia University
Coauthors: Andras Nemethi, Jennifer Hom
A link is called an L-space link if all sufficiently large surgeries along it are L-spaces. It is well known that the Heegaard Floer homology of L-space knots have rank 0 or 1 at each Alexander grading. However, for L-space links with many components the homology usually has bigger ranks and a rich structure. I will describe the homology for algebraic and cable links, following joint works with Andras Nemethi and Jen Hom. The key technical tool is a spectral sequence converging to link homology.
Date received: February 7, 2015
On topology in biology
by
Paul C Kainen
Department of Mathematics and Statistics, Georgetown University.
A brief history of the idea of topology in biology is given, including work of Needham and Rashevsky, Thom, Zeeman, and Arnold, Cozzarelli, White, and Sumners, and some recent developments.
Date received: February 17, 2015
Outer-surface drawings
by
Paul C Kainen
Department of Mathematics and Statistics, Georgetown University
We consider drawings of graphs in surfaces where some closed disk in the surface intersects the graph only in its vertices and only in the boundary of the disk. Basic properties of such drawings are derived and relations to ordinary, "Mobius", and generalized book embeddings are discussed. Related to work of the author (Util. Math/Congr. Num. 1990), Archdeacon et al. (Ars Comb. 1998), Ding et al. (JCT, 2000), Bon et al. (J. Molec. Biol. 2008), and Reidys et al. (Bioinformatics, 2011).
Date received: February 17, 2015
Application of tangle theory to DNA topology
by
Soojeong Kim
Yonsei University, South Korea
Coauthors: Isabel K. Darcy
During biological processes, enzyme (protein) interaction with DNA can change the topology of DNA which results in knotted or linked DNA. The topology of DNA can sometimes be determined by biological methods. However, it is a difficult and laborious process which often doesn't work. Thus tangle analysis was introduced to study various enzyme(protein) actions mathematically.
A tangle is a 3-dimensionalball with strings properly embedded in it. In a tangle model, we assume the protein complex as a 3-dimensional ball and the DNA segments bound by protein as strings embedded inside the ball. When a protein binds to DNA at n-sites, then the DNA-protein complex can be modeled by an n-string tangle.
In the early 1990's, C. Ernst and D. Sumners used 2-string tangle to analyze the conformation of DNA segments within the Tn3 and Phase lamda proteins based on N. Cozzarelli's experiments. In late 2000's, a 3-string tangle analysis of DNA topology within Mu-protein is introducedby I. Darcy et. al. Their work is motivated by S. Pathaniaet. al.'s difference experiment of Mu-transpososome. Recently, the author and I. Darcy developed 4-string tangle analysis of DNA-protein complexes and this theory is generalized to n-string tangle analysis.
Date received: February 22, 2015
Odd torsion in the Khovanov homology of semi-adequate links
by
Adam Lowrance
Vassar College
Coauthors: Radmila Sazdanovic
Shumakovitch showed that the Khovanov homology of an alternating link does not contain odd torsion. We adapt his proof to show that chromatic graph cohomology does not contain odd torsion. Then we use a partial isomorphism between chromatic graph cohomology and Khovanov homology to show that semi-adequate links do not contain odd torsion in certain gradings.
Date received: February 25, 2015
A conjecture on amphicheiral knots
by
Vajira Manathunga
University of Tennessee, Knoxville,TN
Detecting chirality is one of main question in knot theory. For this end, various methods have been developed in the past. The polynomial invariants like Jones, HOMFLYPT, Kauffman are the most prominent methods among them. In 2006, James Conant found that for every natural number n, a certain polynomial in the coefficient of the Conway polynomial is a primitive integer-valued degree n Vassiliev invariant, which named as pcn. It is conjectured that pc4n \mod 2 vanishes on all amphicheiral knots. Another way to formulate this conjecture is, if K is an amphicheiral knot then there is a polynomial F such that C(z)C(iz)C(z2)=F2 Where F ∈ Z4(z2) , C(z) is the Conway polynomial of knot K and i = √{-1}. By work of Kawauchi and Hartley it can be easily shown that this is true for all negative and strong positive amphicheiral knots. However the conjecture still remain unsolved for positive amphicheiral knots (not strong). In this talk we summarize our work on this conjecture.
Date received: February 12, 2015
A Short History of Non-cyclic Knot Theory
by
Kenneth A. Perko, Jr.
325 Old Army Road, Scarsdale, New York 10583
This expository paper illustrates some classical non-cyclic milestones in mathematical knot theory, proving (in part) Riley's 1971 conjecture about coverings of the Kinoshita/Terasaka and Conway knots. It also exhibits new, improved diagrams of another eponymous pair of knots.
Date received: March 4, 2015
Finite-type invariants for virtual knots extending Goussarov-Polyak-Viro invariants
by
Nicolas Petit
Dartmouth College
Kauffman defined a notion of finite-type invariant for virtual knots. Around the same time, Goussarov-Polyak-Viro gave a different definition. It's been known that every GPV finite-type invariant extends to an invariant according to the Kauffman definition. We show in this talk that the space of such extensions is trivial in the unframed case and one-dimensional in the framed case.
Date received: February 26, 2015
Virtual Legenedrian Isotopy
by
R. Sadykov
Dartmouth College
Coauthors: V. Chernov
An elementary stabilization of a Legendrian knot L in the spherical cotangent bundle of a surface M is a surgery that results in attaching a handle to M along two discs away from the image in M of the projection of the knot L. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. A class of virtual Legendrian isotopy is called a virtual Legendrian knot.
We study virtual Legendrian knots and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and Chernov: two Legendrian knots in the spherical cotangent bundle of the 2-sphere that are isotopic as virtual Legendrian knots must be Legendrian isotopic.
Date received: March 6, 2015
Satoh & Rourkes welded knots
by
Jonathan Schneider
UIC Department of Mathematics
In 2001, Satoh found a topological invariant of welded knots, namely, a ribbon torus knot developed from a diagram of the welded knot. In 2008, Rourke strengthened this model by incorporating the toruss fiber structure, making it a complete invariant of welded knots. We explore the relationship between Rourke & Satohs invariants and consider certain generalizations.
Date received: February 20, 2015
On a conjecture of Louis H. Kauffman on alternative and pseudoalternating links
by
Marithania SilveroCasanova
Universidad de Sevilla, Spain and Indiana University
In 1983 Louis Kauffman introduced the family of alternative links, as a generalization of alternating links. It is known that alternative links are pseudoalternating. Kauffman conjectured the converse. In this talk we show that both families are equal in the particular case of knots of genus one. However, Kauffman's Conjecture does not hold in general, as we also show by finding two counterexamples. In the way we will deal with the intermediate family of homogeneous links, introduced by Peter Cromwell; the techniques used here allow us to give an explicit characterization of homogeneous links of genus 1. It was here, at GWU, 2 years ago when I presented the problem and my initial work on it.
Paper reference: arXiv:1402.4599
Date received: February 25, 2015
Bipartite communities
by
Matthew Yancey
Institute for Defense Analysis / Center for Computing Science
A recent trend in data-mining is to find communities in a graph. Generally speaking, a community of a graph is a vertex set such that the number of edges contained entirely inside the set is "significantly more than expected." These communities are then used to describe families of proteins in protein-protein interaction networks, among other applications. Community detection is known to be NP-hard; there are several methods to find an approximate solution with rigorous bounds. In this talk we will discuss the spectral method to find communties with structure that is almost optimal.
We present a new goal in community detection: to find good bipartite communities. A bipartite community is a pair of disjoint vertex sets S, S' such that the number of edges with one endpoint in S and the other endpoint in S' is "significantly more than expected." We claim that this additional structure is natural to some applications of community detection. In fact, using other terminology, they have already been used in two different studies on distinct biological networks. We will show how the spectral methods for classical community detection can be generalized to finding bipartite communities. Additionally, we will present how the algorithm performs on public-source data sets.
Paper reference: arXiv:1412.5666
Date received: February 24, 2015
Copyright © 2015 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.