A mathematical preprint assigns an unknotting number of 3 to the named 11- and 12-crossing knots in one result and to a named group of 13-crossing knots in another. The paper also reports a broader target set of 194 alternating knots with 13 or fewer crossings.
The conclusions are not exact for every group. One listed set is narrowed to an unknotting number of either 3 or 4, without deciding which value applies to each knot. For another group, the paper says that if the unknotting number is 2, the two crossings involved must have the same sign.
How the two-change test works
The stated research question is whether a framework built from d-invariants and an obstruction theorem can determine unknotting numbers for the targeted alternating knots. The main analysis applies Ozsváth and Szabó d-invariants together with Brendan Owens’ obstruction theorem. In this setting, the d-invariants are used as correction terms in a compatibility test for a proposed two-crossing configuration.
Owens’ theorem considers a proposed unknotting with p positive and n negative crossing changes. The conditions require n to equal half the knot’s signature and p+n to equal 2, along with a positive-definite matrix condition. The calculation therefore tests whether the matrix information and correction terms can fit the same proposed route.
For every group element g, the candidate quantity mQ(g) must be at least the corresponding d-invariant and must match it modulo 2. The computation checks whether a group isomorphism can preserve those requirements. If every possible isomorphism is obstructed, the proposed two-crossing configuration is ruled out.
The paper describes two computational routes. For two-bridge knots, it uses lens-space correction terms. For general alternating knots, it uses positive-definite Goeritz matrices, allowing the obstruction framework to be applied across the listed diagrams.
Closing a narrow gap
A worked example shows how the argument reaches an exact answer. For 12a796, the signature is 4, the signature bound gives an unknotting number of at least 2, and the diagram gives a value of either 2 or 3. The subsequent calculation obstructs all possible isomorphisms and concludes that u(12a796)=3.
The example illustrates the role of the obstruction: it removes the lower possibility from a short list of candidates. That same type of reasoning underlies the exact assignments reported for the listed 11- and 12-crossing knots and for the listed 13-crossing knots.
The paper remains cautious about the unresolved cases. A range of 3 or 4 does not identify the exact value for any particular knot. The same-sign conclusion is also conditional: it says what the two crossing changes would have to look like if a listed knot has unknotting number 2, but it does not establish or rule out that value.
Computations available for checking
The appendix provides SageMath and SnapPy implementations for the described process and points readers to a full notebook at GP26. The supplied analysis says the reported target is 194 alternating knots with 13 or fewer crossings, while the named theorem lists are not explicitly reconciled with that overall total in the available text.
The document is labeled arXiv version 1 and dated 26 Aug 2026. The work is reported as using SHARCNET and Digital Research Alliance of Canada facilities, and each author is reported as partially supported by a named graduate scholarship grant.
Paper data and sources
Original title: Obstructing Unknotting Number Two For Some Alternating Knots
Authors: Justin Gebel, William Prangley
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text