Signature invariants related to the unknotting number

Abstract

New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.

The Levine-Tristram signature: a survey

“…Finally, note that for knots, the lower bound on the unknotting number can be significantly improved upon by using the jumps of the signature function [68]. Other applications of the Levine-Tristram signature to unknotting numbers can be found in [87] (as well as a relation to finite type invariants).…”

“…Secondly, we stress that even though σ L was defined 50 years ago, it continues to be actively studied nowadays. We mention some recent examples: results involving concordance properties of positive knots can be found in [3]; the behavior of σ L under splicing is now understood [20]; the relation between the jumps of σ L and the zeroes of ∆ L has been clarified [35,60]; a diagrammatic interpretation of σ L (inspired by quantum topology) is conjectured in [84]; there is a characterization of the functions that arise as knot signatures [67]; new lower bounds on unknotting numbers have been obtained via σ K [68]; there is a complete description of the ω ∈ S 1 at which σ L is a concordance invariant [77]; and σ L is invariant under topological concordance [81].…”