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Mathematics > Geometric Topology

arXiv:2509.18456 (math)
[Submitted on 22 Sep 2025 (v1), last revised 24 Sep 2025 (this
version, v2)]

Title:A Fast, Strong, Topologically Meaningful and Fun Knot Invariant

Authors:Dror Bar-Natan, Roland van der Veen
View a PDF of the paper titled A Fast, Strong, Topologically
Meaningful and Fun Knot Invariant, by Dror Bar-Natan and Roland van
der Veen
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    Abstract:In this paper we discuss a pair of polynomial knot
    invariants $\Theta=(\Delta,\theta)$ which is:
    * Theoretically and practically fast: $\Theta$ can be computed in
    polynomial time. We can compute it in full on random knots with
    over 300 crossings, and its evaluation at simple rational numbers
    on random knots with over 600 crossings.
    * Strong: Its separation power is much greater than the
    hyperbolic volume, the HOMFLY-PT polynomial and Khovanov homology
    (taken together) on knots with up to 15 crossings (while being
    computable on much larger knots).
    * Topologically meaningful: It likely gives a genus bound, and
    there are reasons to hope that it would do more.
    * Fun: Scroll to Figures 1.1-1.4, 3.1, and 6.2.
    $\Delta$ is merely the Alexander polynomial. $\theta$ is almost
    certainly equal to an invariant that was studied extensively by
    Ohtsuki, continuing Rozansky, Kricker, and Garoufalidis. Yet our
    formulas, proofs, and programs are much simpler and enable its
    computation even on very large knots.

Comments:    34 pages
Subjects:    Geometric Topology (math.GT); Quantum Algebra (math.QA)
MSC classes: Primary 57K14, secondary 16T99
Cite as:     arXiv:2509.18456 [math.GT]
             (or arXiv:2509.18456v2 [math.GT] for this version)
             https://doi.org/10.48550/arXiv.2509.18456
             Focus to learn more
             arXiv-issued DOI via DataCite

Submission history

From: Dror Bar-Natan [view email]
[v1] Mon, 22 Sep 2025 22:25:40 UTC (25,663 KB)
[v2] Wed, 24 Sep 2025 12:31:09 UTC (25,663 KB)
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