Timeline for On the total curvature of a knot
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 19, 2016 at 17:22 | comment | added | user21349 | Somewhat related and possibly of interest: Gonzalez and Maddocks, "Global curvature, thickness, and the ideal shapes of knots," Proc. Natl. Acad. Sci. USA 96 (1999) 4769, ma.utexas.edu/users/og/PUBLICATIONS/paper_GlobCurv.pdf | |
| Aug 19, 2016 at 15:16 | comment | added | Zhiyun Cheng | If the fundamental group has $n$ generators, then the bridge number (=crookedness) is at least $n$, which implies the minimal curvature is at least $2\pi n$ (Theorem 4.7 in Milnor's paper). | |
| Aug 19, 2016 at 8:33 | comment | added | Liviu Nicolaescu | www3.nd.edu/~lnicolae/knot-curv.pdf | |
| Aug 19, 2016 at 4:54 | comment | added | Chaitanya | @LiviuNicolaescu : Could you provide a reference for this fact ? It seems very interesting and reminds me of the Cauchy-Crofton formula. Thanks for the response. | |
| Aug 19, 2016 at 4:47 | comment | added | Chaitanya | @RyanBudney : I also believe that crookedness in fact equals n, the number of generators. | |
| Aug 18, 2016 at 10:33 | comment | added | Liviu Nicolaescu | The crookedness as defined by Milnor is the expected number of the restriction to the knot of a random linear function on $\mathbb{R}^3$, where the randomness is uniform in terms of their unit normal defining a linear function. | |
| Aug 18, 2016 at 5:59 | comment | added | Ryan Budney | See mathoverflow.net/questions/32245/… I believe your question is essentially answered there although you need to relate crossing number to minimal number of generators. | |
| Aug 18, 2016 at 5:41 | history | asked | Chaitanya | CC BY-SA 3.0 |