Knots and braids in dimensions 3 and 3.5

Abstract: We describe equivariant sl(2) and sl(3) homology for links in the solid torus, identified with the thickened annulus, via foam evaluation and universal construction. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams that may intersect the distinguished line. Intersection points, called anchor points, contribute additional terms to the foam evaluation. State spaces and link homology carry additional gradings coming from anchor points. I will describe our evaluation formula and the resulting annular homology theories, as well as relations to other constructions. This is joint work with Mikhail Khovanov. 

Abstract:  Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially  over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true.  In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold.

Abstract: Link Floer homology is a powerful link invariant taking value in the category of graded vector spaces. Given such an invariant there are two natural questions one might ask; "Which links have the same link Floer homology?" and "Which vector spaces can arise as the link Floer homology of a link?". This talk addresses several special cases of these questions, and is based on joint work in progress with Subhankar Dey.

What can the failure of the integer lattice to approximate an ellipse tell us about certain 3-manifolds and the knots inside them? This talk will answer that question by introducing lattice homology and knot lattice homology, which are homology theories developed to compute Heegaard Floer homology and knot Floer homology of, respectively, links of normal surface singularities and generalized algebraic knots. After providing the foundation for the three-manifold invariant, lattice homology, in part I, the second video discusses knot lattice homology and outlines proof of its invariance.

Abstract: A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cyclic cover of a knot in the 3-sphere give many examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. This is joint work with Ahmad Issa.

Abstract: This talk builds up to defining a version of Khovanov-Rozansky sl(n) homology via filtered matrix factorizations, which carries information about smooth knot concordance.

Abstract: A cobordism between links induces a map between the Khovanov homologies of the links. The Khovanov map induced by a split cobordism is determined entirely by the individual components of the cobordism. What about non-split cobordisms between split links? For such cobordisms, we will show that the induced Khovanov map does not detect linking information between distinct components. In particular, closed components can be "pulled off". Our main tool will come from lifting Batson and Seed's link invariant to cobordisms and showing that it cannot "see" linking information. This result is used to show that a strongly homotopy ribbon concordance induces an injective map on Khovanov Homology, generalising a result of Levine and Zemke. Joint work with Adam Levine. 

The knot concordance group can be contextualized as organizing problems about 3- and 4-dimensional spaces and the relationships between them. Every 3-manifold is surgery on some link, not necessarily a knot, and thus it is natural to ask about such a group for links. In 1988, Le Dimet constructed the string link concordance groups and in 1998, Habegger and Lin precisely characterized these groups as quotients of the link concordance sets using a group action. Notably, the knot concordance group is abelian while, for each n, the string link concordance group on n strands is non-abelian as it contains the pure braid group on n strands as a subgroup. In this talk, I will discuss my result that even the quotient of each string link concordance group by its pure braid subgroup is non-abelian.

Abstract: A thickened surfaces has many nontrivial covers. It is natural to ask how these covers can be used to study links in the thickened surface; we introduce one way of doing so.

In the first half of the talk we describe this new construction, that takes the form of a theory of parity, as introduced by Manturov. The construction is applicable to arbitrary (compact orientable) surfaces; in the second half of the talk we restrict to the case links in the thickened annulus, and consider potential applications to braids and concordance homomorphisms. Specifically, we define a new conjugacy class invariant of braids, and a way to associate a braid in the m-strand braid group to a braid in the n-strand braid group, for m a divisor of n.

We pose a question an affirmative answer to which would yield a proof of Hedden's conjecture for braid closures and winding number 0 patterns. An affirmative answer to this question would also reduce Hedden's conjecture in nonzero windwing number to the unit winding number case

Abstract: In the first talk, we discuss link cobordisms (e.g., knotted surfaces, slice disks, Seifert surfaces, concordances) and the maps they induce on Khovanov homology. Such induced maps are invariant, up to sign, under boundary-preserving isotopy of the link cobordism. In the second talk, we use this invariance to obstruct the boundary-preserving isotopy between pairs of slice disks by showing that they induce distinct maps on Khovanov homology.

This talk is based on joint work with Kyle Hayden and joint work with Jonah Swann.