| (ↄ) | Dror Bar-Natan: AcademicPensieve: Projects: | About Recent Changes This Month Random |
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--1.4 and 3.1.
<< 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 | MatTheta3 | source | Wed 13 Aug 2025 10:07:59 | Wed 13 Aug 2025 10:16:49 | Roland's notebook on Mat/Weave knots. |
| 5 | Stats13 | source | Tue 29 Jul 2025 04:50:38 | Wed 30 Jul 2025 05:08:33 | The separation powers of various invariants. |
| 6 | Stats14 | source | Wed 30 Jul 2025 04:24:25 | Wed 30 Jul 2025 06:04:36 | The separation powers of various invariants. |
| 7 | Stats | source | Sun 20 Jul 2025 15:29:22 | Sun 3 Aug 2025 13:38:14 | The separation powers of various invariants. |
| 8 | Theta-Antisymmetric | source | Fri 6 Dec 2024 23:32:12 | Fri 6 Dec 2024 23:54:43 | |
| 9 | Theta | source | Mon 16 Sep 2024 11:34:48 | Thu 14 Aug 2025 04:41:22 | 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. |
| 10 | WeaveKnots | source | Wed 13 Aug 2025 10:16:56 | Thu 14 Aug 2025 06:31:45 | Weave knots computations. |
<< Links >>
- Studied Fri 20 Dec 2024 15:10:08: The two-loop knot invariant Theta.
<< Images >>
