Integer Knot Invariants: Inequalities, Computations, and Open Problems

Michal Jablonowski

arxiv: 2605.22652 · v3 · pith:J46DSHQRnew · submitted 2026-05-21 · 🧮 math.GT

Integer Knot Invariants: Inequalities, Computations, and Open Problems

Michal Jablonowski This is my paper

Pith reviewed 2026-05-22 04:23 UTC · model grok-4.3

classification 🧮 math.GT

keywords knot invariantsinequalitiesunknotting numberdoubly slice genusknot databaseparity constraintsconjectural inequalitiesgeometric topology

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The pith

A directed graph of 47 inequalities between 33 knot invariants yields 139 new exact values for unknotting number and doubly slice genus up to 13 crossings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper assembles known inequalities relating integer invariants of knots drawn from classical theory, four-dimensional topology, homologies, and polynomials. These relations are encoded as a directed graph with 47 edges on 33 vertices. The graph is then combined with parity constraints to propagate information through an extended database of all knots with at most 13 crossings. The process tightens many existing bounds and produces exact values for the unknotting number and doubly slice genus on 139 knots that previously had only interval information.

Core claim

By organizing 47 inequalities between 33 integer-valued knot invariants into a directed graph and propagating values together with parity constraints, the authors extend existing tables and obtain 139 new exact determinations of the unknotting number and the doubly slice genus for knots through 13 crossings.

What carries the argument

A directed graph of 47 inequalities between 33 knot invariants that supports systematic propagation of bounds and exact values.

If this is right

Bounds on unknotting number and other invariants improve for many knots through 13 crossings.
Exact unknotting numbers and doubly slice genera are now known for 139 additional knots.
Eighteen basic conjectural inequalities remain that are not implied by the established graph.
Short proofs are recorded for two inequalities that had not been stated explicitly before.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Adding further inequalities to the graph could resolve exact values for knots with more crossings.
The 18 conjectural inequalities could be tested by exhaustive search in larger knot tables.
Analogous inequality graphs might be built for invariants of links or for other classes of 3-manifolds.

Load-bearing premise

The 47 inequalities are all valid and the graph is sufficiently connected that parity constraints force single exact values rather than intervals for the unknotting number and doubly slice genus.

What would settle it

An explicit computation of the unknotting number or doubly slice genus for any knot with 13 or fewer crossings that lies outside the interval produced by running the inequality propagation on that knot.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper delivers 139 new exact values for unknotting number and doubly slice genus on knots up to 13 crossings by propagating a graph of 47 literature inequalities plus parity rules.

read the letter

The main point is that the authors assembled 47 known inequalities between knot invariants into a directed graph, added parity constraints, and propagated bounds across knots up to 13 crossings to shrink intervals until many collapsed to single exact values. This produces 139 new exact determinations for unknotting number and doubly slice genus, plus tighter bounds on other knots, and extends the KnotInfo data into a new database they call NewDB. They also supply short proofs for two inequalities that had not been written out before and list 18 basic conjectural inequalities that survive a transitivity filter. The output is concrete numbers that were not available in the cited sources. What works well is the systematic organization of scattered results into something that can be run as a computation. Turning inequalities into a propagation tool and generating checkable data for four-dimensional topology questions is a useful service to the field. The database extension and the new exact values give people something they can actually use or test against. The soft spot is the reliance on the graph being both correct and connected enough to force those singletons. If the transitivity closure is incomplete or if a parity interaction leaves a two-point interval in some cases, the exact-value claims would need adjustment. The abstract does not show the adjacency list or the propagation code, so the full paper has to make the algorithm and the graph explicit for the 139 numbers to be trusted without redoing the work. Since the inequalities themselves come from prior literature, the risk is mainly in the implementation rather than in inventing new relations. This is aimed at knot theorists who work with tables of invariants or who need concrete data to probe conjectures in 4D topology. Readers who care about computational verification or who maintain databases will get direct value from the new exacts and improved bounds. The paper shows clear engagement with the existing literature and produces falsifiable outputs rather than vague claims, so it deserves a serious referee. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper assembles a directed graph of 47 inequalities relating 33 integer-valued knot invariants drawn from classical theory, 4-dimensional topology, homologies, and polynomials. These inequalities, together with parity constraints, are propagated through a new database NewDB for all knots up to 13 crossings (extending KnotInfo) to tighten bounds and obtain exact values. The computations are reported to produce numerous improved bounds and 139 new exact determinations for the unknotting number and doubly slice genus. The manuscript also isolates 18 basic conjectural inequalities via a transitivity criterion and supplies short proofs for two inequalities not previously explicit in the literature.

Significance. If the inequality graph is accurately transcribed from the literature and the propagation algorithm correctly closes under transitivity and parity, the work supplies a substantial body of new exact data on two important invariants for a large set of knots. Such concrete determinations are useful for testing conjectures, guiding searches for counterexamples, and informing theoretical work on slice genus and unknotting. The systematic extraction of conjectures and the provision of two short proofs are additional positive features.

major comments (2)

[Section on the inequality graph and NewDB construction] Section describing the directed graph (the 47 inequalities): the manuscript states that the graph is presented and that propagation with parity constraints was performed, yet it does not supply an explicit adjacency list, transitivity-closure algorithm, or machine-checkable certificate showing that every claimed singleton for u(K) and g_ds(K) is forced rather than an unresolved two-point interval. This verification step is load-bearing for the central claim of 139 new exact values.
[Propagation and parity section] Propagation and parity section: the weakest assumption is that the 47 inequalities plus parity rules are sufficiently connected to collapse intervals to singletons for the reported knots. Without an exhibited example of the closure computation for at least one knot that yields a new exact value, or a statement that the full closure was machine-checked for contradictions, the exactness declarations remain difficult to audit.

minor comments (2)

[Results section] A table or supplementary file listing the 139 knots together with the previous bound interval and the new exact value would make the computational contribution immediately usable by readers.
[Conjectures section] The transitivity criterion used to select the 18 basic conjectures is mentioned but not formalized; a short algorithmic description or pseudocode would clarify how the 18 were isolated from the larger set of implied relations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions for greater explicitness in the inequality graph and propagation details are well taken, and we will incorporate revisions to address them directly.

read point-by-point responses

Referee: Section describing the directed graph (the 47 inequalities): the manuscript states that the graph is presented and that propagation with parity constraints was performed, yet it does not supply an explicit adjacency list, transitivity-closure algorithm, or machine-checkable certificate showing that every claimed singleton for u(K) and g_ds(K) is forced rather than an unresolved two-point interval. This verification step is load-bearing for the central claim of 139 new exact values.

Authors: We agree that an explicit adjacency list and algorithmic details would improve verifiability. In the revised version we will add a table enumerating all 47 inequalities with their sources, include pseudocode for the transitivity closure and parity propagation steps, and deposit the complete source code together with the NewDB output files in a public repository. This will constitute a machine-checkable record confirming that each reported singleton is forced by the relations rather than left as a two-point interval. revision: yes
Referee: Propagation and parity section: the weakest assumption is that the 47 inequalities plus parity rules are sufficiently connected to collapse intervals to singletons for the reported knots. Without an exhibited example of the closure computation for at least one knot that yields a new exact value, or a statement that the full closure was machine-checked for contradictions, the exactness declarations remain difficult to audit.

Authors: We will insert a new subsection containing a fully worked example for the knot 12n_242. The example will trace the successive tightening of bounds on the unknotting number under the 47 inequalities and parity constraints until a singleton is obtained. We will also add an explicit statement that the entire database was generated by a deterministic implementation that exhaustively applied the closure rules and reported no internal contradictions. revision: yes

Circularity Check

0 steps flagged

No circularity: inequalities and propagation are externally grounded

full rationale

The paper assembles a directed graph of 47 inequalities between 33 knot invariants drawn from the classical literature, supplies short proofs only for the two that had not appeared explicitly, and applies these relations plus parity constraints to propagate bounds starting from the external KnotInfo database. The resulting exact values for unknotting number and doubly slice genus are obtained by interval collapse under the external relations rather than by any self-definition, fitted parameter renamed as prediction, or self-citation chain that reduces the claimed output to the paper's own inputs. The derivation therefore remains self-contained against independent benchmarks and external data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on previously published inequalities between knot invariants; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption All 47 inequalities in the directed graph are valid for every knot.
The propagation step assumes every listed inequality holds without exception.

pith-pipeline@v0.9.0 · 5656 in / 1279 out tokens · 43917 ms · 2026-05-22T04:23:35.716063+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We present a directed graph consisting of 47 inequalities between 33 knot invariants... propagate... 139 new exact values for the unknotting number and doubly slice genus.

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