Claremont Topology Seminar

Founded 1989

2015-16 Schedule

Fall, 2015

Date	Speaker	Title and Abstract
Tuesday Sept 8, 2015 3:00 pm	Organizational Meeting	Meet at Some Crust Bakery for organizational meeting.
Tuesday Sept 15, 2015 3:00 pm	Jim Hoste Pitzer College	Title: Involutory quandles of Montesinos Links I will show how to compute the involutory quandle of the (1/2, 1/2, p/q;e)-Montesinos links and discuss some of he properties of these quandles. This is part of a larger project to classify all links with finite n-quandles. This is joint work with Patrick Shanahan.
Tuesday Sept 22, 2015 3:00 pm	Sam Nelson Claremnt McKenna College	Title: Biquandle brackets Abstract: Given a finite biquandle X and a commutative ring with identity R, we define an algebraic structure known as a biquandle bracket. Biquandle brackets can be used to define a family of knot and link invariants known as quantum enhancements which include biquandle cocycle invariants and skein polynomials such as the Alexander, Jones and HOMFLYpt polynomials as special cases. As an application we will see a new skein invariant which is not determined by the knot group, the knot quandle or the HOMFLYpt polynomial.
Tuesday Sept 29, 2015 3:00 pm	Satyan Devadoss Williams College	Title: Origami Folding and Evolutionary Trees Abstract: In the past 25 years, origami has seen a tremendous explosion, in the arts, the sciences, and in technology. The mathematical theory of origami, in many ways, is at its infancy. There is a simple relationship between origami folds and geometric trees, obtained simply by looking at the crease lines of a piece of folded polygonal paper. In genetics, such trees play an important role in capturing the evolutionary process of species. We try to show a natural map between these worlds, of spaces of polygons and spaces of metric trees, and ask some foundational questions about this map. The heavy lifting of our work is done by an analogous version of a beautiful rigidity result of Cauchy from 1813.
Tuesday Oct 6, 2015 3:00 pm	Caitlin Leverson Duke University	Title: Legendrian knots and links Abstract: qGiven a plane field $dz-xdy$ in $\mathbb{R}^3$. A Legendrian knot is a knot which at every point is tangent to the plane at that point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in $\mathbb{R}^3$, $J^1(S^1)$, and $\#^k(S^1\times S^2)$ as well a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants. No knowledge of Legendrian knots will be assumed though some knowledge of basic knot theory would be useful.
Tuesday Oct 13, 2015 3:00 pm	Jeremy Toulisse University of Southern California	Title: Anti-de Sitter geometry and Teichmüller theory. Abstract: In the nineties, G. Mess discovered deep connections between anti-de Sitter (AdS) geometry and the theory of hyperbolic surfaces. In particular, there exists an equivalence between maximal surface in AdS space-time and minimal Lagrangian map between hyperbolic surfaces. In this talk, we will explain this equivalence and extend it to manifolds with cone singularities.
Tuesday Oct 20, 2015 3:00 pm	No Meeting	Fall Break
Tuesday Oct 27, 2015 3:00 pm	Kenji Kozai UC Berkeley	Title: Regenerating hyperbolic structures from Sol Abstract: Suppose f is a pseudo-Anosov homeomorphism of a surface S, and that M is the mapping torus of f. Thurston's double-limit theorem implies that M admits a hyperbolic structure. However, M also admits a more natural singular Sol structure coming from the invariant measured foliations on S induced by f. When S is a punctured torus, Heusener-Porti-Suarez showed that the Sol structure can be deformed to nearby hyperbolic cone structures on M. By considering these geometries as special cases of projective structures, the result extends to mapping tori of surfaces of higher genus, provided that the pseudo-Anosov map f satisfies some additional properties.
Tuesday Nov 3, 2015 12:15-1:10 pm	Alissa Crans Loyola Marymount College	Title: Hom Quandles Abstract: Analogous to the case for groups, the collection of quandle homomorphisms, Hom(Q, X), has no natural quandle structure. However, if X is an abelian quandle, then the hom set does become a quandle with the obvious pointwise operation. We will consider examples and investigate properties of this hom quandle.
Tuesday Nov 10, 2015 3:00 pm	Faramarz Vafaee California Institute of Technology	Title: A slicing obstruction from the 10/8 theorem Abstract: A knot K in S^3 is smoothly slice if it bounds a disk that is smoothly embedded in the four-ball. From Furuta's 10/8 theorem, we derive a smooth slicing obstruction for knots in S^3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots (i.e. a knot in S^3 which bounds a locally flat disk in D^4) which are not smoothly slice. This work is joint with Andrew Donald.
Tuesday Nov 17, 2015 3:00 pm	Diana Hubbard Boston College	Title: The hunt for effective transverse invariants Abstract: A central aim of modern contact geometry is to find effective invariants of transverse knots, that is, invariants that give more information than classical invariants. In this talk I will describe the background of this problem, explain why a fruitful way to understand transverse knots is via braids, and outline some progress towards defining an effective invariant in the setting of Khovanov homology. This work is joint with Adam Saltz.
Tuesday Nov 24, 2015 3:00 pm	Kyle Chapman UCSB	Title: Ergodicity in the space of polygonal Knots Abstract: Equilateral polygonal knots are often used as a model for molecules and polymers. Additionally, one can use information about equilateral polygonal knots to gather insight into the space of smooth tame knots. Because of these facts, it is often useful to be able to randomly sample the space of equilateral polygonal knots. This has posed a problem, as until recently, the primary methods of random generation have been used without knowing some basic facts about the validity of the method. We will discuss what it means for a random generation method to be ergodic, and what methods have actually been shown to be so.
Tuesday Dec 1, 2015 3:00 pm	Julie Bergner UC Riverside	Title: Fixed points of group actions on unitary partition complexes Abstract: The poset of nontrivial partitions of a finite set can be geometrically realized to obtain a topological pace equipped with an action of the symmetric group. This space and its fixed point spaces by actions of subgroups of the symmetric group has been investigated in a series of papers by Arone, Dwyer, and Lesh and is expected to lead to a new proof of the Whitehead Conjecture. In joint work with Joachimi, Lesh, Stojanoska, and Wickelgren, we consider instead nontrivial decompositions of a finite dimensional complex vector space into orthogonal subspaces, which form a topological poset. We investigate its geometric realization and the fixed point spaces by actions of subgroups of the unitary group.
Tuesday Dec 8, 2015 3:00 pm	Andy Manion UCLA	Title: An overview of categorification and bordered Heegaard Floer homology Abstract: I will give a survey of some of the history of TQFT-type invariants for links, 3-manifolds, and 4-manifolds, directed toward some recent work relating Heegaard Floer homology and algebraic categorifications. I will start with some background focusing on Heegaard Floer homology and Khovanov homology, and recall some known and conjectural links between these theories. Then I will give a brief exposition of the bordered Heegaard Floer homology of Lipshitz, Ozsvath, and Thurston, focusing on similarities between this theory and the definitions of algebraic invariants like Khovanov homology. If time permits, I will briefly discuss a new formulation of bordered Heegaard Floer homology for knots, due to Ozsvath and Szabo in forthcoming work. This theory offers the possibility of bridging the gap between analytic Heegaard Floer invariants and categorified representation-theoretic invariants, including hopefully new connections with Khovanov homology.

Spring 2016 Schedule

Date	Speaker	Title and Abstract
Tuesday Jan 26, 2016	Moshe Cohen Technion	Title: Random 2-bridge Chebyshev Billiard Table Diagrams Abstract: The study of random knotting is mostly experimental with difficulty in defining good probability spaces. Koseleff and Pecker show that all knots can be parametrized by Chebyshev polynomials in three dimensions. These long knots can be realized as trajectories on billiard table diagrams. We use this knot diagram model to study random knot diagrams by flipping a coin at each 4-valent vertex of the trajectory. We truncate this model to study 2-bridge knots together with the unknot. We give the exact probability of a knot arising in this model. Furthermore, we give the exact probability of obtaining a knot with crossing number c. This is joint work with Sunder Ram Krishnan and Chaim Even-Zohar.
Thursday Jan 28, 2016	Colin Adams Williams College	Title: Upper Bounds on Hyperbolic Volume of Links
Tuesday Feb 2, 2016	Tengren Zhang California Institute of Technology	Title: Degeneration of Hitchin representations Abstract: The Hitchin representations are a generalization of Fuchsian representations to the case where the target is a higher rank Lie group. Together, they form a connected component of the corresponding character variety, much like how Teichmuller space is a connected component of the PSL(2,R) character variety. I will describe an analog of the Fenchel-Nielsen coordinates on the Hitchin component, and then use these coordinates to define a large family of deformations in the Hitchin component called "internal sequences". Then, I will explain some geometric properties of these internal sequences, which allows us to conclude some structural similarities and differences between the higher Hitchin components and Teichmuller space.
Tuesday Feb 9, 2016	Mimi Tsuruga UC Davis	Title: Some snags in sphere recognition Abstract: Given a $d$-dimensional simplicial complex $K$, can we determine whether $K$ is a PL-sphere? For $d=3$, Rubinstein found a deterministic algorithm which has since been improved [Thompson, Jaco and Rubinstein] and implemented [Burton]. In 2011, Schleimer showed that this decision problem is in the class NP, followed by Kuperberg and Hass who show that it is also co-NP (assuming the Reimann hypothesis). And that's just dimension 3. In dimension $d \ge 5$, the problem is undecidable [Novikov] and unknown in $d=4$. In practice, however, there are heuristic algorithms that can recognize a given complex to be a sphere (easily) in many situations. We will outline the heuristics implemented in polymake and discuss its limitations as well as some topologically interesting observations that we encountered in our experiments. Joint work with Michael Joswig and Frank H. Lutz.
Tuesday Feb 16, 2016	Jozef Przytycki George Washington University	Title: Knot Theory: from Fox 3-colorings to Yang-Baxter homology Abstract: We start from the short introduction to Knot Theory from the historical perspective, mentioning J. B. Listing work of 1847 and spending some time on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings. In the second part of the talk I will describe how Fox work was generalized to distributive colorings (rack and quandle) and eventually in the work of Jones and Turaev to link invariants via Yang-Baxter operators. Here the importance of statistical mechanics to topology will be mentioned. Finally I will describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology of Fox colorings) and generalize it to homology of Yang-Baxter operators. We speculate, with supporting evidence, on cocycle invariants of knots coming from Yang-Baxter homology. No deep knowledge of Knot Theory, homological algebra or statistical mechanics is assumed, I will work from basic principles.
Tuesday Feb 23, 2016
Tuesday Mar 1, 2016
Tuesday Mar 8, 2016	Hugh Howards Wake Forrest University	Title: Every graph has an embedding in $S^3$ containing no composite knot. Abstract: In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$ there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$. For example, every embedding of the complete graph on 7 vertices must contain a knot as a subgraph inside of it. Recent results showed that as the graphs get more complicated (for example complete graphs on more vertices) they contain more and more complicated knots. In this talk we will look at Flapan and Howards' contrasting result that every graph has an embedding in $S^3$ that contains no composite knots. Undergraduates are most welcome at the talk and no advanced background in topology is necessary.
Tuesday Mar 15, 2016 3:00 pm	No Meeting	Spring Break
Tuesday Mar 22, 2016	Matthew Stoffregen UCLA	Title: The Seiberg-Witten Equations and Homology Cobordism Abstract: In 2013, Manolescu introduced Pin(2)-equivariant Seiberg-Witten Floer homology, an invariant of three-manifolds. He then used it to disprove the triangulation conjecture - namely, he showed that in any dimension at least 5, there exist topological manifolds which do not admit a triangulation. In this talk, we'll discuss the Conley Index, an invariant from dynamical systems that is at the basis of Manolescu's construction. The Conley Index, in short, associates a space to a dynamical system, and might be considered as a broad generalization of Morse homology. From there, we'll discuss several applications of this Floer homology theory to questions of cobordisms among three-manifolds. In particular, we use Seiberg-Witten Floer homology to show that there are three-manifolds not homology cobordant to any Seifert-fibered space.
Tuesday Mar 29, 2016	George Mossessian UC Davis	Title: Stabilizing Heegaard Splittings of High-Distance Knots Abstract: Suppose K is a knot in S3 with bridge number n and bridge distance greater than 2n. We show that there are at most (2n choose n) distinct minimal genus Heegaard splittings of S3-N(K). These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If K has bridge distance at least 4n, then two splittings from different families become equivalent only after n-1 stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for K corresponding to these Heegaard surfaces.
Tuesday Apr 5, 2016
Tuesday Apr 12, 2016
Tuesday Apr 19, 2016	Katherine Raoux Brandeis University	Title: Rationally null-homologous knots and rational Seifert surfaces Abstract: Seifert surfaces have long been a useful tool for defining invariants of knots in the 3-sphere. More generally, Seifert surfaces exist for any null-homologous knot. However, in a general 3-manifold, not every knot bounds a surface since the first homology group may be non-trivial. On the other hand, if a knot represents a torsion homology class, some multiple of the knot will bound a surface. Such a surface is called a rational Seifert surface. I will explain this construction in detail and give examples. In addition, I will show how certain classical knot invariants, such as the genus and self-linking number, can be defined and studied for this more general class of knots. If time permits, I will also discuss how to define tau-invariants from Heegaard Floer homology for this class knots.
Tuesday Apr 26, 2016	Biji Wong Brandeis University	Title: Seiberg-Witten & Turaev torsion invariants of 3-orbifolds Abstract: We construct a torsion invariant of 3-orbifolds with singular set a link. It generalizes the Turaev torsion invariant of 3-manifolds, and gives more information than Baldridge's Seiberg-Witten orbifold invariant. Furthermore, when the singular set is a nullhomologous knot, the invariant not only recovers the Turaev torsion invariant of the underlying 3-manifold, but also detects the isotropy group associated to the knot, and the Turaev torsion invariant of the exterior.
Tuesday May 3, 2016	Anastasiia Tsvietkova UC Davis	Title: Volume of links and the colored Jones polynomial Abstract. Since quantum invariants were introduced into knot theory, there has been a strong interest in relating them to the intrinsic geometry of a link complement. This is for example reflected in the Volume Conjecture, which claims that the hyperbolic volume of a link complement in 3-sphere is determined by the colored Jones polynomial. In the work of M. Lackenby, and of I. Agol and D. Thurston, an upper bound for volume of a hyperbolic link complement in terms of the number of twists of a link diagram is obtained. We will discuss how to refine this bound, and how to generalize it from hyperbolic to simplicial volume of links. We will also show how to express the refined bound in terms of the three first and three last coefficients of the colored Jones polynomial for alternating links. The talk is based on joint work with O. Dasbach.