Integer Knot Invariants: Inequalities, Computations, and Open Problems

Abstract

We study how inequalities among integer knot invariants combine to yield new exact values and sharper constraints. A directed network of forty-seven established inequalities among thirty-three invariants is combined with parity conditions, alternating evaluations, known ribbon status, and interval propagation for prime knots through thirteen crossings. We prove that the propagation is sound, terminating, and order independent. Under a stated source-bound hypothesis, the resulting database reports, as improvements over the source intervals, exact unknotting numbers for thirty-six thirteen-crossing knots and exact doubly slice genera for seventy-two thirteen-crossing knots. Both lists are given explicitly, and we distinguish rows derived only from genus data from those inheriting an imported unknotting bound. Our main theoretical result is a leading-coefficient obstruction to the depth of ordinary oriented binary skein trees. Its specialisation to the Conway polynomial yields two simpler tests using columns already present: a leading-coefficient test and, as its weakest case, a degree-drop test. Combined with previously computed upper bounds, the obstruction determines eleven skein depths exactly; two of these are not detected by the Conway specialisation. We also formulate nine candidate comparisons not implied by the network, prove family cases and structural restrictions for eight of them, and show that the ninth is equivalent to the smooth Slice--Ribbon Conjecture.