Knot cobordisms, bridge index, and torsion in Floer homology

Abstract

Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for Tp,q and Tp,q#T p,q , respectively. We also show that the bridge index of Tp,q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show that knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

Knot cobordism and Lee's perturbation of Khovanov homology

“…, where Kh t is Lee's perturbation of Khovanov homology. Similar results are obtained in the context of knot Floer homology(see [12] [5]).…”

Critical point counts in knot cobordisms: abelian and metacyclic invariants

“…For the torus knot T 2,3 we have br(T 2,3 ) = 2 and it is elementary to see that b(T 2,3 ) = 2. In fact, in [18] it is shown that for torus knots, b(K) = br(K). Yet there are still basic examples that are unresolved: for K = nT 2,3 we have br(K) = n + 1; is it true that b(nT 2,3 ) = n + 1?…”

Ribbon Homology Cobordisms and Link Floer Homology

“…Recall that the fusion number Fus(K) of a ribbon knot K in S 3 is the minimal number of bands in a handle decomposition of ribbon concordance C from the unknot U to K in S 3 × [0, 1]. By [JMZ20], the torsion order of K in S 3 provides a lower bound for the fusion number of K. There are a few possible generalizations we will consider.…”

“…Let K be a nullhomologous knot in a 3-manifold Y . Define the torsion order of K in Y to be the quantity [JMZ20] use the the torsion order of knots in S 3 to give bounds on many topological invariants of knots, including the fusion number, the bridge index, and the cobordism distance. We prove an analogue of [JMZ20, Theorem 1.2] in the ribbon homology cobordism setting.…”