Integer Knot Invariants: Inequalities, Computations, and Open Problems

TL;DR

This paper explores inequalities between integer-valued knot invariants, constructs a comprehensive database for knots up to 13 crossings, and improves bounds on unknotting number and doubly slice genus, while proposing new conjectures.

Contribution

It introduces a directed graph of inequalities, extends knot invariant data, and formulates conjectural inequalities with a systematic approach.

Findings

Produced a database for knots up to 13 crossings extending KnotInfo.

Improved bounds and determined 139 new exact values for unknotting number and doubly slice genus.

Formulated 18 new conjectural inequalities and proved two existing inequalities.

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 47 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 18 basic "interesting" conjectures not implied by the remaining relations. In addition, we record short proofs of the two inequalities which do not seem to have appeared explicitly in the literature.