Dartmouth Topology Seminar
Fall 2023–Spring 2024
Thursdays 10:00-11:00 AM EDT
307 Kemeny Hall
Note: Special meeting times are marked in red.

* Dates marked with an asterisk correspond to a Zoom talk.
The Zoom ID is 870 912 2782; ask Vladimir Chernov for the password.
Date Speaker Title
Feb 29 Mike Miller
University of Vermont
Feb 8 Oleg Lazarev
UMass Boston
Nov 16 * Sergey Melikhov
Steklov Mathematical Institute of the Russian Academy of Science, Moscow, Russia
On the analogues of Cochran's derived invariants for links of linking number 1
Oct 26 Ivan Dynnikov
Steklov Mathematical Institute of the Russian Academy of Science and Moscow State University, Moscow, Russia
Rectangular diagrams of taut foliations in knot complements
Oct 12 Matt Hogancamp
Northeastern University
Khovanov homology and handleslides
Sep 28 Matthew Harper
UC Riverside
Seifert-Torres Formulas for the Alexander Polynomial of Links from Quantum sl2
Sep 14 * Evgeniy Scepin
Steklov Mathematical Institute of the Russian Academy of Science, Moscow, Russia
Application of topology to optical recognition of handwritten characters

February 29, 2023: Mike Miller "TBA"

Abstract: TBA

February 8, 2023: Oleg Lazarev "TBA"

Abstract: TBA

November 16, 2023: Sergey Melikhov " On the analogues of Cochran's derived invariants for links of linking number 1"

Abstract: Let L=(K_1,K_2) be a two-component smooth link in S^3. In the case lk(L)=0, S. Kojima and M. Yamasaki (1979) introduced a useful invariant \eta_L, namely, the generating function of the linking numbers between a lift of a zero pushoff of K_2 and the various lifts of K_2 itself in the infinite cyclic cover of S^3-K_1. (Here the "linking number" has the usual meaning when K_1 is the unknot, and in general it is defined using that every 1-cycle in \widetilde{S^3-K_1} becomes a boundary upon being multiplied by some Laurent polynomial.) T. Cochran (1985) discovered that \eta_L is equivalent by a change of variable to a power series whose coefficients \beta^i(L) have a simple and beautiful geometric description in terms of iterated intersections of Seifert surfaces. It further turns out that these integers \beta^i(L) are in fact some of the coefficients of the power series \nabla_L(u,v,w)/\nabla_{K_1}(u), where \nabla_{K_1} is the Conway polynomial and \nabla_L is the so-called two-variable Conway polynomial. This observation of mine (2003) was originally a consequence of a theorem of J. T. Jin (1988), who expressed \eta_L in terms of the Alexander polynomials of L and K_1 by a clever argument involving many surgeries of S^3. But recently I found a simpler proof of this fact, in terms of Seifert matrices.
Cochran's derived invariants \beta^i(L), being coefficients of the power series \nabla_L(u,v,w)/\nabla_{K_1}(u), are defined for two-component smooth links of any linking number. But in the case lk(L)\ne 0 their geometry was a complete mystery until recently. I will discuss a theorem which sheds some light on their geometry in the case lk(L)=1.
One reason to be concerned about the geometry of \beta^i(L) in the case lk(L)=1 is that these invariants have been used in the recent progress on Rolfsen's problem: "Is every knot isotopic to the unknot? In particular, is the Bing sling isotopic (=homotopic through embeddings) to the unknot (or, equivalently, to a smooth knot)?". The theorem says that there exists no isotopy from the Bing sling to the unknot that extends to an isotopy of a two-component link with lk=1.

October 26, 2023: Ivan Dynnikov "Rectangular diagrams of taut foliations in knot complements"

Abstract: Taut foliations are an important instrument in low-dimensional topology. In particular, due to the works of W.Thurston and D.Gabai, they can be used to certify knot genus. Jointly with Mikhail Chernavskikh we have worked out a universal way to represent taut foliations in knot complements by using the formalism of rectangular diagrams, and shown that any finite depth taut foliation can be represented in this way. This will be explained in the talk. The work is supported by the Russian Science Foundation under grant 22-11-00299

October 12, 2023: Matt Hogancamp " Khovanov homology and handleslides "

Abstract: If K is a framed knot and X is a link in the solid torus, then the satellite operation produces a link K(X), obtained by embedding X (the "pattern") into a tubular neighborhood of K (the "companion"). In the context of Khovanov homology, the patterns can be thought of as objects of a category called the annular Bar-Natan category (ABN), and the satellite operation defines a functor from ABN to bigraded vector spaces, sending a pattern X to Kh(K(X)). In this talk I will discuss joint work with Dave Rose and Paul Wedrich, in which we construct an object Ω (which we call a "Kirby color") in ABN such that Kh(K(Ω)) is invariant under handleslides. As I will explain, the object Ω encodes the Manolescu-Neithalath 2-handle formula for the sl(2) skein lasagna modules (which was inspirational for our work). Time permitting, I will discuss an intriguing description of the Kirby object in terms of some special braids (positive braid lifts of n-cycles) that we speculate may be more amenable for computation.

September 28, 2023: Matthew Harper "Seifert-Torres Formulas for the Alexander Polynomial of Links from Quantum sl2"

Abstract: In this talk, I will recall how the Alexander polynomial, a classical knot invariant, can be constructed as a quantum invariant from quantum sl2 at a fourth root of unity. I will then discuss the development of a diagrammatic calculus based on further investigation of quantum sl2 representations. Applying this calculus in the context of the Alexander polynomial allows us to compute the invariant for certain families of links using quantum algebraic methods, rather than using methods of classical topology.

September 14, 2023: Evgeniy Scepin "Application of topology to optical recognition of handwritten characters"

Abstract: Every handwritten symbol consists of several lines. These lines may intersect. Therefore, each symbol corresponds to a graph whose vertices correspond to the intersection points of the lines and their ends, and the edges correspond to the line segments enclosed between the vertices. A scanned image of a symbol representing a Boolean matrix is fed to the input of the recognition program. On the image matrix, the lines become "thick", interference also occurs there. The report will tell you how to reduce the image so as to get a correct graph of the symbol, whose edges on the matrix are represented by thin (pixel-thick) lines that preserve the shape of the original ones.