Topology at IU

Seminars During Academic Year 2022-23

Fall 2022

Wednesday September 7, 2022 in SE 105: Daniel Ramras (IUPUI), "Flat connections and the topological Atiyah-Segal map"; Given a closed, aspherical manifold M, the topological Atiyah–Segal is a map of ring spectra relating the topological K-theory of M to a spectrum built from representations of the fundamental group of M. The homotopy fiber of this map can be described in terms of spaces of flat connections on vector bundles over M. Combined with work of Tyler Lawson, this allows one to extract information about spaces of flat connections from homological data regarding spaces of irreducible representations. I’ll explain this set-up, and describe some applications.
Wednesday October 19, 2022 in SE 105: Agniva Roy (GA Tech), "Small caps and embedding rational homology balls in CP^2"; Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP^2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.
Wednesday October 26, 2022 in SE 105: Seppo Niemi-Colvin, "Invariance of Knot Lattice Homology and Homotopy"; Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic complex surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.
Wednesday November 2, 2022: Kai Smith (IU), "Traceless SU(2) Character Varieties of Tangles"; Hedden, Herald, and Kirk initiated a program to study knots by decomposing them into tangles and studying the interaction between certain perturbed moduli spaces associated to each tangle. I will explain this program and how it is related to gauge theory, in particular Kronheimer and Mrowka's singular instanton homology. I will present my own results which lead to computations of these perturbed moduli spaces for many new tangles. These computations shed some light on a proposed knot invariant called pillowcase homology.
Wednesday November 9, 2022 in Zoom Only: Sally Collins (Georgia Tech), "Homology cobordism and knot concordance"; Array
Wednesday November 16, 2022 in SE 105: Taehee Kim (Konkuk University), "Iterated satellite operators and knot concordance"; Hedden and Pinz\'{o}n-Caicedo conjecture that the image of a non-constant satellite operator generates an infinite rank subgroup in the knot concordance group. For a satellite operator P, they also ask a similar question on the P-filtration of the knot concordance group which is obtained by applying P iteratively. In this talk, we will address the conjecture/question and related problems using L^2-signatures. In particular, we will show that for a winding number zero satellite operator P, if the axis has nontrivial Blanchfield self-linking form, then each of the successive quotient groups of the P-filtration has infinite rank. This is joint work with Jae Choon Cha.

Spring 2023

Wednesday January 18, 2023 in Zoom Only: 609 643 9799 Password: derivative: David Boozer (Princeton University), "Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus"; We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,\delta)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,\delta)$ is an isotopy invariant of the tangle diagram. We use $(X,\delta)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.
Wednesday April 19, 2023 in SE 105: Taehee Kim (Konkuk Univ.), "The double slice genus of a knot"; The double slice genus of a knot in the 3-sphere is the minimum genus among unknotted surfaces in the 4-sphere whose intersection with the 3-sphere is the knot. In this talk, we relate the double slice genus of a knot to the second Betti number of a closed 4-manifold into which the zero-surgery on the knot can be embedded preserving first homology. Then, using the relationship and the von-Neumann rho invariants, we give examples of knots with vanishing Casson-Gordon invariants whose double slice genera are arbitrarily large. This joint work with Se-Goo Kim.
Wednesday April 26, 2023 in SE 105: Shawn Williams (IU), "A Higher Order Fox-Milnor Theorem"; Abstract: In a celebrated theorem, Fox and Milnor showed the Alexander polynomial of a slice knot factors as $f(t)f(t^{-1})$. We introduce a family of Alexander modules $\mathcal{A}_n^\mathcal{P}$ that peer more deeply into the knot group than the classical Alexander module. Our main result is a new sequence of Fox-Milnor Theorems for (n+1)-solvable knots.

IU/PU/IUPUI Joint Topology Seminar

Fall 2022: Saturday September 24, 2022: IU/PU/IUPUI Joint Topology Seminar, Sanjana Agarwal (IU) and Sam Nariman (Purdue), TBA