Dror Bar-Natan: AcademicPensieve: Projects: Theta:

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This is the construction / computation page for my joint paper with Roland van der Veen:

A Very Fast, Very Strong, Topologically Meaningful and Fun Knot Invariant.

Paper PDF here: Theta.pdf. Computations here: Theta.nb.

Abstract. In this paper we introduce $\Theta=(\Delta,\theta)$, a pair of polynomial knot invariants 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 gives a genus bound, and there are reasons to hope that it would do more.
Fun: Scroll to Figures 1.1, 1.2, and 3.1.

$\Delta$ is merely the Alexander polynomial. $\theta$ is almost certainly equal to an invariant that was studied extensively by Ohtsuki [Oh], continuing Rozansky, Garoufalidis, and Kricker [GR, Ro1, Ro2, Ro3, Kr]. Yet our formulas, proofs, and programs are much simpler and enable its computation even on very large knots.

<< Subfolders / Shortcuts >>

Archive figs KnotFigs MRP ProgramVariants Sage Snips Vols / Projects: APAI Projects: HigherRank: DunfieldKnots Projects: HigherRank Talks: KnotTheoryCongress-2502 Talks: Toronto-241030

<< PDF / TXT >>

PPDemo.pdf Theta4Rolfsen.pdf Theta_Journal.pdf Theta.pdf / recycling.txt

<< Mathematica Notebooks >>

	Notebook (.pdf)	Source (.nb)	Created	Last Modified	Summary
1	index	source	Tue 10 Sep 2024 13:37:57	Sun 2 Feb 2025 11:40:59	This is the index file for the Theta project.
2	Make	source	Sun 17 Nov 2024 12:56:59	Fri 1 Aug 2025 03:28:59
3	MatTheta2	source	Wed 30 Apr 2025 15:38:50	Wed 30 Apr 2025 16:42:31
4	Stats13	source	Tue 29 Jul 2025 04:50:38	Wed 30 Jul 2025 05:08:33	The separation powers of various invariants.
5	Stats14	source	Wed 30 Jul 2025 04:24:25	Wed 30 Jul 2025 06:04:36	The separation powers of various invariants.
6	Stats	source	Sun 20 Jul 2025 15:29:22	Tue 29 Jul 2025 04:41:23	The separation powers of various invariants.
7	Theta-Antisymmetric	source	Fri 6 Dec 2024 23:32:12	Fri 6 Dec 2024 23:54:43
8	Theta	source	Mon 16 Sep 2024 11:34:48	Thu 31 Jul 2025 10:39:22	This is the main Mathematica package that goes along with the paper "A Very Fast, Very Strong, Topologically Meaningful and Fun Knot Invariant" by Dror Bar-Natan and Roland van der Veen.