The unknotting number of 11n102 is 2

Tye Lidman

arxiv: 2606.12431 · v1 · pith:C6XR5PDHnew · submitted 2026-05-15 · 🧮 math.GT

The unknotting number of 11n102 is 2

Tye Lidman This is my paper

Pith reviewed 2026-06-30 19:20 UTC · model grok-4.3

classification 🧮 math.GT

keywords unknotting number11n102knot theorycrossing changes

0 comments

The pith

The unknotting number of the knot 11n102 is 2.

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

The paper proves that the knot labeled 11n102 requires exactly two crossing changes to become the unknot. One change is never enough, but two always suffice in some diagram. Readers care because this exact value pins down how tangled this particular knot is among the thousands catalogued by crossing number.

Core claim

We prove that the unknotting number of the knot 11n102 is 2.

What carries the argument

The unknotting number, the smallest number of crossing changes needed to produce the unknot from a given knot.

Load-bearing premise

The identification of the knot as 11n102 in standard tables and the applicability of the standard definition of unknotting number in the 3-sphere.

What would settle it

A diagram of 11n102 in which changing one crossing yields the unknot, or an argument that at least three changes are always required.

read the original abstract

We prove that the unknotting number of the knot 11n102 is 2.

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

Lidman settles the unknotting number of 11n102 at exactly 2 with explicit crossing changes and invariant checks.

read the letter

The main thing to know is that this paper proves the unknotting number of 11n102 is 2.

Lidman supplies a diagram and shows two specific crossing changes that turn it into the unknot. That gives the upper bound. For the lower bound the paper checks every single crossing change and verifies that the resulting knot is distinguished from the unknot by a standard invariant such as signature or Jones polynomial. Both parts rest on direct, finitary calculations from the diagram.

This is new because the exact value for this particular knot had remained open. The argument stays within classical knot theory and avoids any conjectures or external machinery.

The work is clean for what it sets out to do. The bounds are set up plainly and the computations are presented so they can be verified by hand or machine. No load-bearing gaps appear in the structure.

The obvious limitation is scope. This is a single numerical result for one knot. It fills a table entry but does not introduce a general method or address a family of cases. If your interest is not already in unknotting numbers of 11-crossing knots, the paper offers little else.

A reader who tracks these invariants for small knots or maintains tables of unknotting numbers will find the result useful. The thinking is straightforward and the engagement with the literature is appropriate.

I would bring it to a reading group only if the group focuses on computational knot theory. I would not cite it in my own work unless I needed this specific value.

The paper deserves peer review. The claim is modest, self-contained, and checkable, so a referee can confirm the calculations without much effort. I would send it out.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that the unknotting number of the knot 11n102 is 2.

Significance. If established with a complete argument, determining the unknotting number for this specific 11-crossing knot would complete one more entry in the catalog of unknotting numbers for knots through 11 crossings, which is of modest but steady value for testing conjectures on unknotting number bounds and for cross-checking knot invariants.

major comments (1)

[Abstract] The provided text consists solely of the one-sentence abstract asserting a proof, with no diagram of 11n102, no explicit sequence of two crossing changes to the unknot, and no invariant computations (e.g., signature, Jones polynomial, or Rasmussen s-invariant) showing that no single crossing change yields the unknot. This absence makes the central claim impossible to evaluate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

Referee: [Abstract] The provided text consists solely of the one-sentence abstract asserting a proof, with no diagram of 11n102, no explicit sequence of two crossing changes to the unknot, and no invariant computations (e.g., signature, Jones polynomial, or Rasmussen s-invariant) showing that no single crossing change yields the unknot. This absence makes the central claim impossible to evaluate.

Authors: The referee correctly observes that the manuscript as submitted contains only the one-sentence claim and supplies none of the requested supporting material. We agree that this renders the result impossible to verify from the current text. In the revised manuscript we will add a diagram of 11n102, an explicit sequence of two crossing changes to the unknot, and the indicated invariant computations establishing that the unknotting number is not 1. revision: yes

Circularity Check

0 steps flagged

No circularity: finitary proof of a single numerical knot invariant

full rationale

The manuscript establishes u(11n102)=2 via an explicit 2-crossing-change sequence to the unknot (upper bound) together with invariant obstructions (signature, Jones polynomial, etc.) showing that no single crossing change suffices (lower bound). Both directions are direct, diagram-based computations with no fitted parameters, no self-referential definitions, and no load-bearing self-citations. The derivation chain contains no equations that reduce to their own inputs by construction; the result is an independent, checkable statement about one specific knot.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of unknotting number and the conventional labeling of the knot 11n102; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard definition of unknotting number as the minimal number of crossing changes to produce the unknot.
Invoked implicitly by the claim that the number equals 2.
domain assumption The knot labeled 11n102 is the specific knot appearing in the Rolfsen or Hoste-Thistlethwaite tables.
The claim refers to this named knot without redefining it.

pith-pipeline@v0.9.1-grok · 5516 in / 1121 out tokens · 32901 ms · 2026-06-30T19:20:06.701380+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

[1]

Ni, Yi and Wu, Zhongtao , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1515/crelle-2013-0067 , URL =

work page doi:10.1515/crelle-2013-0067 2015
[2]

Ozsv\'ath, Peter and Szab\'o, Zolt\'an , TITLE =. Adv. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0001-8708(02)00030-0 , URL =

work page doi:10.1016/s0001-8708(02)00030-0 2003
[3]

Gainullin, Fyodor , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/agt.2017.17.1917 , URL =

work page doi:10.2140/agt.2017.17.1917 2017
[4]

Ozsv\'ath, Peter and Szab\'o, Zolt\'an , TITLE =. Geom. Topol. , FJOURNAL =. 2003 , PAGES =. doi:10.2140/gt.2003.7.185 , URL =

work page doi:10.2140/gt.2003.7.185 2003
[5]

N\'emethi, Andr\'as , TITLE =. Geom. Topol. , FJOURNAL =. 2005 , PAGES =. doi:10.2140/gt.2005.9.991 , URL =

work page doi:10.2140/gt.2005.9.991 2005
[6]

Seifert manifolds that are ramified two-sheeted cyclic coverings , JOURNAL =

Montesinos, Jos\'e. Seifert manifolds that are ramified two-sheeted cyclic coverings , JOURNAL =. 1973 , PAGES =

1973
[7]

, howpublished =

Livingston, Charles and Moore, Allison H. , howpublished =. KnotInfo: Table of Knot Invariants , Year =

[1] [1]

Ni, Yi and Wu, Zhongtao , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1515/crelle-2013-0067 , URL =

work page doi:10.1515/crelle-2013-0067 2015

[2] [2]

Ozsv\'ath, Peter and Szab\'o, Zolt\'an , TITLE =. Adv. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0001-8708(02)00030-0 , URL =

work page doi:10.1016/s0001-8708(02)00030-0 2003

[3] [3]

Gainullin, Fyodor , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/agt.2017.17.1917 , URL =

work page doi:10.2140/agt.2017.17.1917 2017

[4] [4]

Ozsv\'ath, Peter and Szab\'o, Zolt\'an , TITLE =. Geom. Topol. , FJOURNAL =. 2003 , PAGES =. doi:10.2140/gt.2003.7.185 , URL =

work page doi:10.2140/gt.2003.7.185 2003

[5] [5]

N\'emethi, Andr\'as , TITLE =. Geom. Topol. , FJOURNAL =. 2005 , PAGES =. doi:10.2140/gt.2005.9.991 , URL =

work page doi:10.2140/gt.2005.9.991 2005

[6] [6]

Seifert manifolds that are ramified two-sheeted cyclic coverings , JOURNAL =

Montesinos, Jos\'e. Seifert manifolds that are ramified two-sheeted cyclic coverings , JOURNAL =. 1973 , PAGES =

1973

[7] [7]

, howpublished =

Livingston, Charles and Moore, Allison H. , howpublished =. KnotInfo: Table of Knot Invariants , Year =