Dartmouth Topology Seminar
Fall 2020–Spring 2021
Thursdays 11am-noon or 1:30-2:30 PM EDT
201 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Apr 29, 1:30pm Vladimir Nezhinski
(St Petersburg State University)
Isotopy invariants of space disk-ribbon graphs.
Apr. 22, 1:30pm Roman Golovko
(Charles University Prague)
Subloose Legendrian tori from Bohr-Sommerfeld covers of monotone Lagrangian tori
Apr. 15, 1:30pm Juanita Pinzon-Caicedo
(University of Notre Dame)
Toroidal integer homology spheres have irreducible SU(2)-representations.
Apr. 8, 1:30pm Patricia Cahn
(Smith College)
Branched Covers of Trisected 4-Manifolds
Feb. 11, 1:30pm Josh Howie
(UC Davis)
Alternating genera of torus knots
Feb. 4, 1:30pm Rima Chatterjee
Knots and links in overtwisted contact manifolds
Jan. 14, 11am Rustam Sadykov
Lusternik-Schnirelmann theory of 4-manifolds
Jan. 7, 11am Mikhail Tyomkin
(Higher School of Economics, Moscow)
Alexander polynomial of string links and Gassner matrix

April 29, 2021: Vladimir Nezhinski "Isotopy invariants of space disk-ribbon graphs."

Abstract: The main goal of the talk is to reduce the problem of isotopy classification of space graphs, equipped with an additional structure - a framing, a skeleton and a marked point, to the problem of isotopy classification of tangles.

April 22, 2021: Roman Golovko "Subloose Legendrian tori from Bohr-Sommerfeld covers of monotone Lagrangian tori"

Abstract: By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside CP^n for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms.
In addition, we will discuss the conjecture relating superpotential of an embedded monotone Lagrangian two-torus in CP2 with the augmentation polynomial of the Legendrian lift of its canonical threefold Bohr-Sommerfeld cover. This is joint work in progress with Georgios Dimitroglou Rizell.

April 15, 2021: Juanita Pinzon-Caicedo " Toroidal integer homology spheres have irreducible SU(2)-representations."

Abstract: Toroidal integer homology spheres have irreducible SU(2)-representations. Abstract: The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. This is joint work with Tye Lidman and Raphael Zentner.

April 8, 2021: Patricia Cahn "Branched Covers of Trisected 4-Manifolds"

Abstract: A well-known theorem due to Hilden and Montesinos states that every closed, oriented 3-manifold is a 3-fold branched cover of $S^3$, branched along a knot. Piergallini later proved an analogous result in dimension 4: every closed, oriented 4-manifold is a 4-fold branched cover of $S^4$, branched along an immersed surface. A trisection of an oriented, closed, 4-manifold X is a decomposition of X into three 4-dimensional handlebodies, analogous to Heegaard splittings in dimension 3. We prove that if one of the three handlebodies in a trisected 4-manifold X is a 4-ball, then X is a 3-fold (rather than 4-fold) branched cover of $S^4$. The branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. This is joint work with Ryan Blair, Alexandra Kjuchukova, and Jeffrey Meier.

February 11, 2021: Josh Howie "Alternating genera of torus knots"

Abstract: The alternating genus of a knot is the minimum genus of a surface onto which the knot has an alternating diagram satisfying certain conditions. Very little is currently known about this knot invariant. We study spanning surfaces for knots, and define a related invariant from the extremal spanning surfaces. This gives a lower bound on the alternating genus and can be calculated exactly for torus knots. We find the first examples of knots where the alternating genus is equal to n for each n>2, and classify all toroidally alternating torus knots.

February 4, 2021: Rima Chatterjee "Knots and links in overtwisted contact manifolds"

Abstract: Knot theory associated to overtwisted manifolds is less explored. There are two types of knots/links in an overtwisted manifold namely loose and non-loose. These knots are different than the knots in tight manifolds in many ways. In this talk, I'll start with an overview of these knots/links and discuss my recent results on classifying loose null-homologous links. Next, I'll talk about an invariant named support genus of knots and links and show that this invariant vanishes for loose links. I'll end with some interesting open questions and future work directions.

January 14, 2021: Rustam Sadykov "Lusternik-Schnirelmann theory of 4-manifolds"

Abstract: The Lusternik-Schnirelmann category of a topological space X is a minimal number of open sets U_i in a cover of X such that each set U_i is contractible in X. I will discuss various versions of the Lusternik-Schnirelmann category of 4-manifolds. In particular, I will discuss the relation of the Lusternik-Schnirlemann theory of 4-manifolds to Gay-Kirby trisections.
This is a joint work with Stanislav Trunov.

January 7, 2021: Mikhail Tyomkin "Alexander polynomial of string links and Gassner matrix"

Abstract: String link is a certain generalization of a pure braid, where we allow string to go upwards. One can define Alexander polynomial of a string link in the same manner as for the usual honest closed link in a 3-sphere. On the other hand, given string link L one can consider its closure \hat{L}, which is a closed link. We will discuss a formula which relates Alexander polynomials of L and \hat{L}. Namely, these two polynomials differ by a determinant of a Gassner matrix --- a certain invariant of a string link, which is interesting by itself. We assume no prerequisites.