Dartmouth Topology Seminar
Fall 2019–Spring 2020
Thursday 2-3 PM
201 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Feb. 13 Andy Manion
Jan. 16 Gage Martin
(Boston College)
Nov. 14 Siddhi Krishna
(Boston College)
Taut Foliations, Positive 3-Braids, and the L-Space Conjecture
Nov. 7 James Cornish
(Boston College)
Growth of Heegaard Floer Homology in Branched Covers
Oct. 31 Ivan Dynnikov
(Steklov Mathematical Institute)
Distinsguishing Legendrian and transverse knots
Oct. 24 Egor Shelukhin
(University of Montreal)
Symplectic cohomology and a conjecture of Viterbo
Oct. 3 Bülent Tosun
(University of Alabama)
Stein domains in complex 2-plane with prescribed boundary
Sept. 26 Andrei Maliutin
(Steklov Mathematical Institute)
Decomposition of prime knots

February 13, 2020: Andy Manion "TBA"


January 16, 2020: Gage Martin "TBA"


November 14, 2019: Siddhi Krishna "Taut Foliations, Positive 3-Braids, and the L-Space Conjecture"

Abstract: The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsváth and Szabó. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll build taut foliations for manifolds obtained by surgery on positive 3-braid closures. Our theorem provides the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. As an example, we'll construct taut foliations in every non-L-space obtained by surgery along the P(-2,3,7) pretzel knot. No background in Floer homology or foliation theories will be assumed.

November 7, 2019: James Cornish "Growth of Heegaard Floer Homology in Branched Covers"

Abstract: If $\Sigma_n(K)$ is the n-fold cyclic branched cover of $K$, what happens to the rank of HF or HFK as n goes to infinity? This talk will start with a brief overview of what happens to the size of $H_1(\Sigma_n(K))$ as n goes to infinity. We will then discuss what is known about the Heegaard Floer case as well as a few results to at least bound what happens in this case.

October 31, 2019: Ivan Dynnikov "Distinsguishing Legendrian and transverse knots "

Abstract: A smooth knot (or link) K in the three-space $\mathbb R^3$ is called Legendrian if the restriction of the 1-form $\alpha=x\,dy+dz$ on K vanishes, where x,y,z are the standard coordinates in $\mathbb R^3$. If $\alpha|_K$ is everywhere non-vanishing on K, then K is called transverse.
Classification of Legendrian and transverse knots up to respectively Legendrian and transverse isotopy is an important unsolved problem of contact topology. A number of useful invariants have been constructed in the literature, but there are still small complexity examples in which the existing methods do not suffice to decide whether or not the given Legendrian (or transverse) knots are equivalent.
We propose a totally new approach to the equivalence problem for Legendrian and transverse knots, which allows one to practically distinguish between non-equivalent Legendrain (or transverse) knots in small complexity cases, and gives rise to a complete algorithmic solution in the general case.
The talk is based on joint works (recent and in progress) with Maxim Prasolov and Vladimir Shastin.

October 24, 2019: Egor Shelukhin "Symplectic cohomology and a conjecture of Viterbo"

Abstract: We discuss a recent proof of a conjecture of Viterbo regarding uniform bounds on the spectral norm of exact Lagrangian submanifolds inside a fixed cotangent disk bundle of the n-dimensional torus. It uses methods of symplectic cohomology, and works more generally for a new class of smooth manifolds.

October 3, 2019: Bülent Tosun "Stein domains in complex 2-plane with prescribed boundary"

Abstract: In this talk, I would like to discuss the question of "which integral homology spheres can be embedded into complex 2-plane as the boundaries of Stein submanifolds". This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that: a Brieskorn homology sphere, with either orientation, cannot be embedded into complex 2-plane as the boundary of a Stein submanifold. In this talk, I will provide partial progress (some completed work as well as report on some recent work in progress) towards resolving this conjecture.

September 26, 2019: Andrei Maliutin "Decomposition of prime knots"

Abstract: In mathematics, various objects admit canonical decompositions into prime components: we have the fundamental theorem of arithmetic for integers, the fundamental theorem of algebra for polynomials, the Jordan normal form, the ergodic decomposition, etc., etc. For 3-manifolds we have a two-level decomposition: the prime decomposition (the Kneser-Milnor theorem) and the JSJ decomposition. A similar two-level decomposition is known for knots: Schubert's theorem on decomposition into primes and the JSJ decomposition for knot complements. It turns out that the knots (but not the links) have also a third-level decomposition: each prime knot has a canonical decomposition into two-strand Conway irreducible tangles. We will discuss this decomposition into tangles. An interesting application of this decomposition is a complete classification of mutant knots.