Timeline for Looking through a bunch of links for unlinks?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 29, 2020 at 14:58 | vote | accept | user101010 | ||
| Jul 28, 2020 at 6:49 | comment | added | dvitek | @user101010 If you need to ensure you've detected all of the unlinks and no more, I'm not sure if SnapPy can guarantee that. (Other people in this thread would know more.) For polynomials this is somewhat harder - once you start hitting 15-16 crossings Thistlethwaite has examples of nontrivial links whose Jones polynomials are those of the appropriate unlinks. However, Khovanov homology detects unlinks (Batson-Seed) and you only have to work over $\mathbb{F}_2$, not integrally. This will be slower than other approaches but may be your only bet for unlink detection with no false positives. | |
| Jul 28, 2020 at 6:47 | comment | added | dvitek | @user101010 In that case computation of knot polynomials is also pretty feasible. Naive algorithms should take time $\mathrm{poly}(n)\cdot 2^n$, whereas more sophisticated algorithms can look like $\mathrm{poly}(n)\cdot 2^{\sqrt{n}}$. See the Knot Atlas page on computations of the Jones polynomial for how to implement this for polynomials satisfying skein relations. I think Cotton Seed has some code floating around under the name knotkit that might implement good algorithms, but you'd have to check. (Caveat: I don't have experience with the relevant parts of SnapPy.) | |
| Jul 27, 2020 at 14:13 | comment | added | user101010 | @dvitek Sure thing - yeah I have diagrams of all of these links and crossing numbers are in the 10-20 range. | |
| Jul 27, 2020 at 3:28 | comment | added | Ian Agol | SnapPy is actually pretty good at detecting unlinks. Just compute the fundamental group, and it can usually see that it is a free group. | |
| Jul 27, 2020 at 1:28 | comment | added | dvitek | It would probably help if you provided (1) how these links are described - e.g., do you have diagrams? triangulations of the complement? - and (2) roughly how complex these descriptions are on average - e.g., average crossing number of the diagrams. | |
| Jul 26, 2020 at 19:58 | answer | added | Sam Nead | timeline score: 3 | |
| Jul 26, 2020 at 17:09 | answer | added | Carl-Fredrik Nyberg Brodda | timeline score: 2 | |
| Jul 26, 2020 at 13:10 | history | asked | user101010 | CC BY-SA 4.0 |