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A Fast, Strong, Topologically Meaningful and Fun Knot Invariant.
Paper PDF here: Theta.pdf. Computations here: Theta.nb.
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 and 3.1.
<< Mathematica Notebooks >>
| Notebook (.pdf) | Source (.nb) | Created | Last Modified | Summary | |
|---|---|---|---|---|---|
| 1 | FiberedKnots | source | Wed 20 Aug 2025 08:02:54 | Thu 28 Aug 2025 03:05:23 | Playing with fibered knots. |
| 2 | index | source | Tue 10 Sep 2024 13:37:57 | Fri 22 Aug 2025 01:20:21 | This is the index file for the Theta project. |
| 3 | KnotGenus | source | Tue 26 Aug 2025 06:11:47 | Tue 26 Aug 2025 10:15:27 | Verifying the genus conjecture. |
| 4 | Make | source | Sun 17 Nov 2024 12:56:59 | Thu 28 Aug 2025 03:26:07 | |
| 5 | MatTheta2 | source | Wed 30 Apr 2025 15:38:50 | Wed 30 Apr 2025 16:42:31 | |
| 6 | MatTheta3 | source | Wed 13 Aug 2025 10:07:59 | Wed 13 Aug 2025 10:16:49 | Roland's notebook on Mat/Weave knots. |
| 7 | OddPretzels2 | source | Mon 25 Aug 2025 05:23:13 | Thu 28 Aug 2025 02:54:22 | |
| 8 | Stats13 | source | Tue 29 Jul 2025 04:50:38 | Wed 30 Jul 2025 05:08:33 | The separation powers of various invariants. |
| 9 | Stats14 | source | Wed 30 Jul 2025 04:24:25 | Wed 30 Jul 2025 06:04:36 | The separation powers of various invariants. |
| 10 | Stats | source | Sun 20 Jul 2025 15:29:22 | Sun 3 Aug 2025 13:38:14 | The separation powers of various invariants. |
| 11 | Theta4Links | source | Tue 19 Aug 2025 07:11:21 | Wed 20 Aug 2025 04:26:07 | |
| 12 | Theta-Antisymmetric | source | Fri 6 Dec 2024 23:32:12 | Fri 6 Dec 2024 23:54:43 | |
| 13 | Theta | source | Mon 16 Sep 2024 11:34:48 | Thu 28 Aug 2025 05:42:32 | 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. |
| 14 | WeaveKnots | source | Wed 13 Aug 2025 10:16:56 | Wed 27 Aug 2025 04:28:02 | Weave knots computations. |
<< Links >>
- Studied Fri 20 Dec 2024 15:10:08: The two-loop knot invariant Theta.
<< Images >>
