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. 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.

• The paper introduces a comprehensive network of 47 inequalities connecting 33 integer knot invariants extracted from diverse aspects of knot theory.
• The paper employs computational techniques via the NewDB database, yielding 139 new exact values and refined bounds for knot invariants in low-crossing knots.
• The paper formulates 50 conjectural inequalities and proves two explicit bounds, paving the way for further algorithmic proofs and deeper topological insights.

Integer Knot Invariants: Structure, Computational Advances, and Conjectural Inequalities

Overview of Integer-Valued Knot Invariants and Their Interrelations

The paper establishes a comprehensive network of inequalities connecting 33 prominent integer-valued knot invariants. These invariants arise from classical knot theory (crossing, bridge, braid indices), four-dimensional topology (slice genus, clasp number), knot homologies (Floer-theoretic, Rasmussen invariant), and polynomial invariants (spans and degrees of Alexander, Jones, HOMFLYPT, and Kauffman polynomials). The author formalizes 47 such inequalities, visualized as a directed graph, where each edge $X \rightarrow Y$ encodes the universal relation $X(K) \geq Y(K)$ for any knot $K$ in $S^3$.

This framework unifies disparate constraints across surface genera, diagrammatic complexity, concordance, skein-theory, and homological constructions. Parity constraints are intertwined, leveraging invariants’ even/odd values in propagation. Notably, two inequalities—linking doubly slice genus with concordance unknotting number and HOMFLYPT z-degree with clasp number—are proved explicitly as they do not follow from classical relations.

Database Construction and Computational Results

To concretize theoretical relations, the author constructs a new database, NewDB, encompassing the 33 invariants for knots up to 13 crossings. The methodology iterates from the KnotInfo resource, expanding to include additional invariants and extracting missing values from literature and direct computation via SageMath. By recursively applying the established inequalities and parity constraints, NewDB yields improved bounds, interval reductions, and a substantive increase in exact values for some invariants.

Quantitative highlights include:

• 139 new exact values for the unknotting number and doubly slice genus.
• For the unknotting number: 36 knots are confirmed to possess $u=2$.
• For the doubly slice genus: 88 knots are confirmed to possess $g_{ds}=4$, and 15 knots have $g_{ds}=6$.

These results provide substantial refinement in the landscape of known knot invariants and extend computational accessibility for subsequent research in low-crossing knots.

Conjectural Inequalities and Open Problems

Systematic analysis of the directed graph and computational data exposes a set of “interesting” inequalities not implied by transitiveness, satisfied universally in NewDB, and realized as equalities and strict inequalities on certain knots. Fifty such conjectures are formulated, from which eighteen fundamental ones are isolated via a minimality criterion.

These conjectures bridge knot invariants in novel ways, for instance:

• $g_{ds}(K) \leq 2u_*(K)$ (doubly slice genus vs. weak ribbon unknotting number)
• $|s(K)| \leq \deg P_z(K)$ (Rasmussen invariant vs. HOMFLYPT polynomial z-degree)
• $a(K)-2 \leq c(K)$ (arc index vs. crossing number, generalizing Nutt's conjecture)

Short proofs for $X(K) \geq Y(K)$0 and $X(K) \geq Y(K)$1 are delivered. The proof of $X(K) \geq Y(K)$2 constructs an orientable surface in $X(K) \geq Y(K)$3 via a concordance and two capping surfaces, yielding the genus bound. $X(K) \geq Y(K)$4 leverages the skein-theoretic tree structure and immersed disks, bounding polynomial degree via clasp-associated resolution.

Implications and Future Directions

The proposed structure of integer knot invariants facilitates streamlined deduction of bounds and interrelations, potentially guiding algorithmic approaches to knot classification and complexity measures. The computational propagation method significantly enhances the precision and completeness of invariants catalogued for low-crossing knots.

Conjectural inequalities suggest deeper topological and algebraic connections, especially between distinct domains: diagrammatic complexity, four-dimensional topology, and knot homology. Their resolution would have implications for concordance class characterization, surface embedding theory, and the computability of polynomial invariants.

Future development may explore:

• Extension to knots with higher crossing numbers and links.
• Connections with quantum invariants and categorified knot polynomials.
• Algorithmic proof or counterexample construction for the BasicConj conjectures.
• Statistical and combinatorial analysis of invariant distributions in large knot databases.

Conclusion

The paper provides a systematic codification of integer-valued knot invariants, their inter-inequalities, and their propagation through computational techniques. New exact values are determined, bounds tightened, and conjectural relations articulated, significantly enriching current understanding. The unified treatment and results lay foundational groundwork for further research into the algebraic, topological, and computational aspects of knot invariants (2605.22652).