Dartmouth Topology Seminar
Fall 2018–Spring 2019
Fall 2018: Thursday 4-5 PM
Winter and Spring 2019: Thursday 2:30-3:30 PM
201 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Apr. 25 Daniel Vitek
(Princeton University)
Apr. 18 Steven Boyer
Apr. 11 Alexander Dranishnikov
(University of Florida)
Apr. 4 Robert Low
(Coventry University)
Mar. 28 Akram Alishahi
Feb. 28 Joshua Sussan
Feb. 21 Mike Wong
Jan. 17 Andrei Maliutin
(Steklov Mathematical Institute)
On the question of genericity of hyperbolic knots
Jan. 10 David Freund
Singular Based Matrices for Virtual 2-Strings
Nov. 1 Ina Petkova
(Dartmouth College)
Tangle Floer homology for non-Floerists
Oct. 11 Samantha Allen
(Dartmouth College)
Nonorientable surfaces bounded by knots
Oct. 4 Shelly Harvey
(Rice University)
A non-discrete metric on the group of topologically slice knots
Sept. 20 Lev Tovstopyat-Nelip
(Boston College)
The transverse invariant and braid dynamics

April 25, 2019: Daniel Vitek "TBA"


April 18, 2019: Steven Boyer "TBA"


April 11, 2019: Alexander Dranishnikov "TBA"


April 4, 2019: Robert Low "TBA"


March 28, 2018: Akram Alishahi "TBA"


February 28, 2018: Joshua Sussan "TBA"


February 21, 2018: Mike Wong "TBA"


January 17, 2018: Andrei Maliutin "On the question of genericity of hyperbolic knots"

Abstract: A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of n or fewer crossings approaches 1 as n approaches infinity. We show that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

January 10, 2018: David Freund "Singular Based Matrices for Virtual 2-Strings"

Abstract: A singular virtual 2-string $\alpha$ is a wedge of two circles on a closed oriented surface. Up to equivalence by virtual homotopy, $\alpha$ can be realized on a canonical surface $\Sigma_\alpha$. We use the homological intersection pairing on $\Sigma_\alpha$ to associate an algebraic object to $\alpha$ called a singular based matrix. In this talk, we show that these objects can be used to distinguish virtual homotopy classes of 2-strings and to compute the virtual Andersen--Mattes--Reshetikhin bracket of families of 2-strings.

November 1, 2018: Ina Petkova "Tangle Floer homology for non-Floerists"

Abstract: A knot is a circle in 3-space. A main problem in knot theory is distinguishing knots (two knots are equivalent if we can continuously deform one into the other). One way to approach this is by studying algebraic "knot invariants" — algebraic objects associated to knots, which do not change as the knot is deformed. In 1928, J. Alexander described a knot invariant, now called the Alexander polynomial. In the early 2000s, Ozsvath and Szabo constructed a powerful refinement of the Alexander polynomial, called knot Floer homology (HFK). Among other properties, it detects the genus, detects fiberedness, and gives a lower bound on the 4-ball genus. The original definition involves counting holomorphic curves in a high-dimensional manifold, and as a result can be hard to compute.

Tangle Floer homology is a new algebraic technique for studying HFK, by cutting a knot into pieces called tangles, and studying the individual pieces and their gluing. One associates a differential graded algebra (DGA) to a sequence of points, and a dg bimodule over the respective DGAs to a tangle with two sets of "ends". Given a decomposition of a knot into tangles, the derived tensor product of the bimodules associated to the pieces recovers the knot Floer homology of the glued knot. After providing a bit of general background, we'll try to sketch out a purely combinatorial definition of this invariant.

October 11, 2018: Samantha Allen "Nonorientable surfaces bounded by knots"

Abstract: The nonorientable 4–genus of a knot K is the minimal first Betti number of a nonorientable surface in B^4 whose boundary is K. Finding the nonorientable 4–genus of a knot can be quite intractable; existing methods exploit the relationship between nonorientable genus and normal Euler number of the nonorientable surface. In this talk, I will give an overview of the interplay between the nonorientable genus and normal Euler number of nonorientable surfaces in B^4. I will define both of these invariants and discuss their computation. In particular, when fixing a knot K, we can ask what pairs of nonorientable genus and normal Euler number are realizable for a surface whose boundary is K. We will see that both classical invariants and Heegaard–Floer invariants can be used towards answering this question.

October 4, 2018: Shelly Harvey "A non-discrete metric on the group of topologically slice knots"

Abstract: Most of the 50-year history of the study of the set of smooth knot concordance classes, C, has focused on its structure as an abelian group. Tim Cochran and I took a different approach, namely we studied C as a metric space (with the slice genus metric) admitting many natural geometric operators. The goal was to give evidence that the knot concordance is a fractal space. However, both of these metrics are integer valued metrics and so induce the discrete topology. Subsequently, with Mark Powell, we defined a family of real valued metrics, called the q-grope metrics, that take values in the real numbers and showed that there are sequences of knots whose q-norms get arbitrarily small for q>1. However, for q>1, this metric vanishes on topologically slice knots (it is really a pseudo metric). In this talk, we define a new metric (called the tower metric) based on a new objects which we called positive and negative towers, using a combination of generalized handles and gropes. This is an interesting metric since it is related to the bipolar filtration, a filtration generalizing work of Gompf and Cochran. Using recent work of Cha and Kim on the non-triviality of the bipolar filtration of the group of topologically slice knots, we show that there are sequences of topologically slice knots whose q-norms get arbitrarily small but are never 0. This work is joint with Tim Cochran, Mark Powell, and Aru Ray.

September 20, 2018: Lev Tovstopyat-Nelip "The transverse invariant and braid dynamics"

Abstract: Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,xi). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the transverse invariant of any braid having fractional Dehn twist coefficient greater than one is non-zero. This generalizes a theorem of Plamenevskaya about classical braid closures.