Timeline for Does this knot invariant distinguish trefoil chiralities?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| May 4, 2012 at 10:47 | history | edited | ARupinski | CC BY-SA 3.0 |
Added update
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| May 4, 2012 at 1:45 | comment | added | ARupinski | Yeah, must have messed up that when I looked it up originally. But all my other knots' chiralities were computed by comparing to the reduced form I did of that one, so the chiralities are at least consistent (so if that one is wrong, all of the $N=8$ ones I calculated below will change chirality too). | |
| May 3, 2012 at 15:36 | comment | added | Dylan Thurston | Isn't the non-trivial knot with $N=7$ a right-handed trefoil, not a left-handed trefoil? Or am I getting some convention backwards? | |
| May 3, 2012 at 11:29 | comment | added | JeffE | Here's a quick proof of the non-intersecting chord result. Denote the moment curve by $\mu(t) = (t, t^2, t^3)$, and consider any four points $\mu(t_1), \mu(t_2), \mu(t_3), \mu(t_4)$ with $t_1 < t_2 < t_3 < t_4$. The volume of the convex hull of these points is $1/6$ the determinant of the matrix whose $i$th row is $[1, t_i, t_i^2, t_i^3]$. This determinant is an antisymmetric polynomial with total degree $6$ in the $t_i$'s, and thus must be $\prod_{i<j} (t_j-t_i) \ne 0$. The same result also holds for one turn of the standard helix $(t, \cos t, \sin t)$ where $0 \le t < 2\pi$. | |
| May 1, 2012 at 2:23 | history | edited | ARupinski | CC BY-SA 3.0 |
Added pictures and background; deleted 8 characters in body; deleted 4 characters in body
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| Apr 30, 2012 at 23:27 | comment | added | ARupinski | Your illustration is close to what I am describing. Hopefully later this evening I will have time to create and upload some pictures of the two trefoils I described above. | |
| Apr 30, 2012 at 23:23 | vote | accept | ARupinski | ||
| Apr 30, 2012 at 18:13 | answer | added | Dylan Thurston | timeline score: 20 | |
| Apr 30, 2012 at 15:45 | comment | added | Joseph O'Rourke | Interesting. I think the figure I drew for mathoverflow.net/questions/95477 is accidentally an illustration? If I am not misinterpreting, then perhaps the relationship to torus knots might be helpful... | |
| Apr 30, 2012 at 12:59 | history | asked | ARupinski | CC BY-SA 3.0 |