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 $T_{p,q}$ and $T_{p,q}\#{\overline{T}}_{p,q}$, respectively. We also show that the bridge index of $T_{p,q}$ 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 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.

arXiv comment21 pages, 7 figures, to appear in the Journal of Topology

References in corpus (13)

• Floer homology and knot complements
• Holomorphic disks and link invariants
• Concordance homomorphisms from knot Floer homology
• Contact structures, sutured Floer homology and TQFT
• Cobordisms of sutured manifolds and the functoriality of link Floer homology
• Link cobordisms and functoriality in link Floer homology
• Scissor equivalence for torus links
• The Bar-Natan homology and unknotting number
• Concordance maps in knot Floer homology
• Contact handles, duality, and sutured Floer homology
• Knot Floer homology and the unknotting number
• Ribbon distance and Khovanov homology
• Tangle Floer homology and cobordisms between tangles

Cited by in corpus (10)

• Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials
• Ribbon cobordisms between lens spaces
• Khovanov homology and rational unknotting
• Non-orientable link cobordisms and torsion order in Floer homologies
• Twist positivity, L-space knots, and concordance
• Ribbon cobordisms as a partial order
• Upper and lower bound on delta-crossing number and tabulation of knots
• On homology planes and contractible $4$-manifolds
• On knots that divide ribbon knotted surfaces
• Ma-Qiu index, presentation distance, and local moves in knot theory