Integer Knot Invariants: Inequalities, Computations, and Open Problems

Abstract

We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 46 inequalities between 33 knot invariants. Using these inequalities together with parity constraints, we construct and propagate a database NewDB, for knots up to 13 crossings, extending data from KnotInfo. The resulting computations produce numerous improvements of known bounds and determine 139 new exact values for the unknotting number and doubly slice genus. We also formulate a collection of conjectural inequalities selected by a systematic transitivity criterion. Among them are 10 basic "interesting" conjectures not implied by the remaining relations.