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 $47$ inequalities between $33$ knot invariants. Propagating known values through this graph for all $12965$ non-trivial prime knots up to $13$ crossings produces numerous strengthened bounds. The current \textsf{NewDB} version records, in particular, the $36$ knots whose unknotting number is pinned to the value $2$ and the $103$ knots whose doubly slice genus is pinned. We retain $10$ transitivity-irredundant candidate inequalities, establish family cases for nine of them---covering alternating, torus, fibered, homogeneous, signature-thin, quasi-alternating and positive knots (the last with equality)---and show that the remaining one is equivalent to the smooth Slice--Ribbon Conjecture. We further verify that the graph is acyclic, that no displayed arrow is a transitive consequence of the others, and that the ten candidates stay independent even when added jointly. We also show that the propagation is confluent, so that the fixed point to which all numerical statements refer is independent of the order in which the rules are applied. Finally, we report a complete structural audit of the distributed workbook against the rule set, including consistency check and a fixed-point verification of the interval propagation.