On the geography and botany of knot Floer homology

Abstract

This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

A note on the knot Floer homology of fibered knots

“…Thus, to prove that ξ is tight it suffices to show that b(ξ) > 1. Theorem 1.8 also provides a simpler solution to the word problem in the mapping class group of a surface with connected boundary than in [HW14]. Indeed, suppose ϕ is a diffeomorphism of Σ which restricts to the identity on ∂Σ.…”

“…As alluded to above, Hedden and Watson used this result in [HW14] to prove that the rank of knot Floer homology detects the trefoil. Though it was not observed in [HW14], their result is also strong enough to show that L-space knots are prime, by an argument similar to ours. We emphasize that Hedden and Watson's proof of Theorem 1.7 is conceptually very different from our proof of Theorem 1.1.…”

Recent advances on the non-coherent band surgery model for site-specific recombination

“…A discussion of this conjecture can be found in [31]. Another nice discussion of some related problems in knot theory and three-manifold topology can be found in Hedden and Watson's article [36,Section 6].…”

The Alexander module, Seifert forms, and categorification

“…Theorem A.1 follows from Theorems A.4, A.11, and A.33, which equate our Z/2Z-gradings gr, gr D , gr A , and gr DA with (a generalization of) Petkova's Z/2Z reduction of the G-set gradings of [17, Chapter 10] and [21, Section 6.5]. Since the G-set gradings are differential gradings (alternatively, type A or DA gradings, in the cases of CFA and CFDA, respectively) and invariant up to the appropriate notion of equivalence ([17, Theorem 10.39] and [21, Theorem 10]), it follows that gr D , gr A , and gr DA are as well.…”