Dartmouth Topology Seminar
Fall 2020–Spring 2021
Thursdays 10-11am EDT
201 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Nov 4 Sergey Maksimenko
Corresponding member of the National Academy of Science of Ukraine, President of the Kiev Mathematical Society, Institute of mathematics of NAS of Ukraine, Kyiv, Ukraine
Combinatorial symplectic symmetries of smooth functions on surfaces
Oct 28 Samuel Tripp,
Dartmouth College
Integral Grid Homology for Links in Lens Spaces
Oct 21 Zachary Winkeler,
Dartmouth College
Multipunctured Khovanov homology
Oct 14 Daniel Alvarez-Gavela,
Legendrian invariants from algebraic K-theory
Oct 7 Misha Tyomkin,
Dartmouth College
Various facets of torsion theory: Whitehead, Reidemeister and Milnor
Sept 30 Misha Tyomkin,
Dartmouth College
Numbers on barcode of a strong Morse function
Sept 23 Emanuele Zappala,
University of Tartu, Estonia
Cocycle invariants of knots and knotted surfaces

November 4, 2021: Sergy Maksimenko "Combinatorial symplectic symmetries of smooth functions on surfaces"

Abstract:Let M be a compact orientable surface equipped with a symplectic form w, P be either the real line of the circle, and f:M→P a smooth function. Let also S(f,w) = {h ∈ Diff(M) | foh=f and h*w=w} be the group of diffeomorphisms of M which mutually preserve f and w, and p:S(f,w)→π0S(f,w) be the natural projection into the group π0S(f,w) of path components of S(f,w). It is shown that for a large class of smooth maps f:M→P, which includes all Morse maps, the homomorphism p admits a section homomorphism s: π0S(f,w)→S(f,w). In other words, in each isotopy class γ of diffeomorphisms mutually preserving f and w one can choose a representative hγ ∈ S(f,w) so that the collection {hγ}γ constitute a subgroup of S(f,w), i.e. hγhγ' = hγ γ'. A variant of such result for non-orientable surfaces is also given. As an application it is also shown that every finite group is a subgroup of S(f,w) for some M, f and w.

October 28, 2021: Samuel Tripp "Integral Grid Homology for Links in Lens Spaces"

Abstract: Baker, Grigsby and Hedden presented a combinatorial description for the knot Flor homology of links in lens spaces. We provide a combinatorial proof of invariance of this grid homology, and adapt this to prove the invariance of the generalization of this grid homology to integral coefficients.

October 21, 2021: Zachary Winkeler "Multipunctured Khovanov homology"

Abstract: Khovanov homology is a knot invariant that categorifies the Jones polynomial. For knots in a thickened annulus, we can define a filtration on the Khovanov chain complex that gives us an invariant called annular Khovanov homology. In this talk, we will discuss a generalized notion of filtration that allows us to construct an analogous invariant for knots in thickened disks with any number of punctures.

October 14, 2021: Daniel Alvarez-Gavela "Legendrian invariants from algebraic K-theory"

Abstract:The theory of generating functions allows one to build Legendrian invariants out of Morse theoretic invariants. While such homological invariants have been extensively studied and compared to pseudo-holomorphic curve invariants, K-theoretic invariants are somewhat more mysterious. In this talk I will discuss K-theoretic torsion invariants for Legendrians in 1-jet spaces, focusing on an example which appeared in previous work of Igusa and Klein, where it was used to provide a picture for a generator of K_3(Z)=Z/48. Joint work with K. Igusa.

October 7, 2021: Misha Tyomkin "Various facets of torsion theory: Whitehead, Reidemeister and Milnor"

Abstract:Torsion is a topological invariant defined algebraically by an appropriate generalization of a notion of determinant. Some of its incarnations are:
* Whitehead torsion is an obstruction for a famous h-cobordism theorem to hold true in the non-simply connected case.
* Reidemeister torsion is an invariant which is able to detect homotopy equivalent, but non-homeomorphic manifolds, such as lens spaces.
* Milnor torsion is a low-dimensional variation of a Reidemeister one, which in some cases equals to the Alexander polynomial of knot.
In this overview talk I will try to touch on these incarnations without delving into proofs or technical details.

September 30, 2021: Misha Tyomkin "Numbers on barcode of a strong Morse function"

Abstract: Morse function f on a manifold M is called strong if all its critical values are pairwise distinct. For a given field F, Barannikov decomposition (a.k.a. barcode) is a canonical pairing of some critical points of such f with neighboring indices. We present a construction which naturally associates a number (i.e. an element of F) to each Barannikov pair (a.k.a. bar in the barcode), defined up to a sign. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f (up to sign). We then proceed to consider homology with twisted coefficients (still in some field F). It is in this setting where the Reidemeister torsion is defined. We construct the twisted barcode and prove that the mentioned product equals to the Reidemeister torsion (provided that the latter is defined); in particular, it's again independent of f. Based on a joint work with Petya Pushkar

September 23, 2021: Emanuele Zappala "Cocycle invariants of knots and knotted surfaces "

Abstract: In this talk I will introduce the notions of racks and quandles, as well as their cohomology. I will discuss the diagrammatic interpretation of these algebraic structures, and show how to use this to construct a partition function that is an invariant of Reidemeister moves. Using the same principles, I will also show that it is possible to obtain an invariant of knotted surfaces in 4-space, defining certain partition functions based on quandle 3-cocycles. These results were introduced in Trans. Amer. Math. Soc. 355 (2003) 3947-3989 by Carter, Jelsovsky, Kamada, Langford and Saito. Then, I will discuss how to generalize the previous results by means of ternary quandle cohomology to get invariants of framed links and compact oriented surfaces with boundary (embedded in 3-space). I will present applications of all these invariants throughout the talk, including a generalization of the cocycle invariants to quantum invariants.