Schedule for: 22w5171 - Using Quantum Invariants to do Interesting Topology

We will look at some new constructions of closed exotic 4-manifolds that can be detected using Ozsvath-Szabo's closed 4-manifold invariants. We build these from the ground up, looking at 3-manifolds with relatively small Heegaard Floer homology groups  on which we can explicitly understand 2-handle cobordism maps and mapping class group actions. This is joint work with Tye Lidman and Lisa Piccirillo.

Reporting on joint work with Roland van der Veen, I'll tell you some stories about $\rho_1$, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. $\rho_1$ was first studied by Rozansky and Overbay, it is dominated by the coloured Jones polynomial (but it isn't lesser!), it has far-reaching generalizations, and I wish I understood it. Further content at http://drorbn.net/oa22

I will survey recent developments that show Khovanov homology to be an effective tool for studying knotted surfaces in the 4-ball. We will take a broad view, including some comparisons with tools from Floer homology and gauge theory, and I will suggest potential directions for further development of these Khovanov-theoretic tools. (This draws on joint work with subsets of {Alan Du, Gary Guth, Sungkyung Kang, Seungwon Kim, Maggie Miller, JungHwan Park, and Isaac Sundberg}.)

In the world of Heegaard Floer theory, the simplest class of 3-manifolds is the class of L-spaces and the simplest class of knots is the class of $(1,1)$-knots. With the objective of understanding the Dehn-surgery landscape of L-spaces, it is natural to restrict to this class of knots. In this context, a theorem due to Greene, Lewallen and Vafaee provides a criterion for the existence of a (non-infinite) surgery slope on any given $(1,1)$-knot that yields an L-space. This criterion is wonderfully simple and, during my MSc, I reproved their theorem for knots that satisfy an additionnal technical assumption by using the immersed curve formalism due to Hanselman, Rasmussen and Watson. In this talk I will describe to you the Heegaard Floer theory of $(1,1)$-knots, where the objects are completely explicit and the computations are combinatorial, and I will tell you about the work I did.

On the (closed) 3-dimensional disk we have a complete classification of smooth (or locally linear) involutions up to conjugation: they are determined by the dimension of their fixed-point set. On the 4-ball, however, the situation is less clear. For example, there are known to be infinitely many distinct involutions with a fixed 2-disk (distinguished by the isotopy class of the disk). In this talk I will discuss work with Wenzhao Chen in which we constructed locally linear involutions whose fixed-point set is an arc by studying the Alexander polynomials of strongly negative amphichiral knots. We suspect that these involutions are not all conjugate, and that a similar construction can produce examples in the smooth category. 

The $T$--genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot. This invariant was introduced by Murakami and Sugishita, who proved that it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their $T$--genus. This can be generalized to provide a $3$--dimensional characterization of the slice genus. Further, we will see that the difference between the $T$--genus and the slice genus can be arbitrarily large, using recent results on the 4D clasp number.

The tau-invariant from Heegaard-Floer homology and the Rasmussen invariants s_c from Khovanov homology (the latter exist for every prime or zero characteristic c) are equal for many families of knots. However, twisted Whitehead doubles are well-suited to distinguish those invariants. The values the Rasmussen invariant s_c takes on twisted Whitehead doubles is determined by another knot invariant theta_c. We will show that theta_2 is a smooth concordance homomorphism that is independent of the s_c. Furthermore, theta_2 determines the value of s_2 not just for twisted Whitehead doubles, but for all satellites with wrapping number two and winding number zero. These proofs rely on the multicurve description for Khovanov homology of four-ended tangles. This talk is based on joint work with Claudius Zibrowius.

Involutive Heegaard Floer homology (HFI), introduced by Hendricks and Manolescu in 2017, incorporates the conjugation action on Ozsváth and Szabó's Heegaard Floer homology to produce a richer 3-manifold invariant. In 2019, Hendricks and Lipshitz introduced Involutive Bordered Floer homology, a modular (i.e. cut-and-paste) version of HFI, and also exhibited a surgery exact triangle for the theory.

Surprising conjectures assert that hyperbolic volumes of 3-manifolds can be recovered from certain quantum invariants when the quantum parameter $q$ has finite order, by considering the asymptotic behavior of these invariants as we let the order of the quantum parameter go to infinity. I will discuss such a conjecture, for diffeomorphisms of surfaces. In particular, I will try to convey how, in this case, the 3-dimensional hyperbolic geometry can emerge from the combinatorics of the relevant quantum invariant.

The Khovanov homology of a link in S^3 has a Lee deformation, which gives rise to Rasmussen's s-invariant which in turn provides topological information on surfaces bounded by the link. When generalizing Khovanov homology to links in other 3-manifolds then, it is desirable to also obtain a Lee-like deformation giving a Rasmussen-like s-invariant giving similar topological information. In this talk I will describe two such situations: for links in connect sums of S^1 x S^2 (joint with Manolescu, Marengon, and Sarkar) and for links in RP^3 (joint work in progress with Manolescu), along with some of the topological applications that follow.

We study symmetric unions in the context of problems regarding knot invariants. Recently, a class of symmetric unions was proposed to construct nontrivial knots with trivial Jones polynomial. We show, however, that such a knot is always trivial and hence this construction cannot be used to answer the open question asking whether the Jones polynomial detects the unknot. We then discuss why symmetric union is a valuable construction to study knot invariants and why our result provides strong evidence for the non-existence of nontrivial knots with trivial Jones polynomial.

We construct a sequence of commutative, finitely presented rings, and link invariants taking values in these rings. We prove that the collection of these invariants determines the fundamental group of the link complement. Within these rings, there is an algebraic obstruction to the existence of any kind of local sequence of tangle transformations, taking the form of a solvability question for a system of linear equations over the ring. These obstructions can theoretically be computed by a Gröbner basis algorithm, which would give us lower bounds on unknotting number and gordian distance. These rings have other interesting properties, such as the fact that if they turn out to be integral domains, then Vassiliev invariants determine the fundamental group. We propose future research towards reducing the complexity of computations in these rings, and discuss relationships with other potentially useful invariants coming from generalized state sum theories which are theoretically but not practically computable.

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.