Talk:Crossing number

Split into separate article

I propose splitting this into two articles on knots and graphs and making this page a disambiguation. Comments? Incidentally, I am planning on expanding the knot theory one into an article probably about the length of the current one. --C S (talk) 13:11, 31 May 2008 (UTC)

Ok with me. They're unrelated topics that each deserve an article; in other cases of this nature we have two articles. My only concern is that WP:MOSDAB#Disambiguation pages with only two entries disrecommends making disambiguation pages with only two entries. —David Eppstein (talk) 16:14, 31 May 2008 (UTC)

I think it only recommends not doing it when there is a primary meaning. Do you think either one of the two is primary? It's not so clear to me. --C S (talk) 01:16, 1 June 2008 (UTC)

Nor to me. Google scholar finds about 1000 articles for "crossing-number knot", about 2000 for "crossing-number graph", and about 4000 for "crossing-number". I wonder what the other 1000 are about... —David Eppstein (talk) 01:26, 1 June 2008 (UTC)

Crossing Number Inequality

Unlike stated in that section, the crossing number inequality apparently has been improved recently: Pach, János; Radoi\v ci\'c, Rado\v s; Tardos, Gábor; Tóth, Géza Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom. 36 (2006), no. 4, 527--552. [[1]] If the paper contains an unconditional (asymptotic) improvement over the known bound, this result should be included. (I haven't read through the paper yet.)Hermel (talk) 13:34, 4 July 2008 (UTC)

They show that when e ≥ (103/6)v , the number of crossings is roughly 1/31.081 e²/v³, improving the previous constant by a little at a big expense in how dense the graph needs to be before the improvement kicks in. (The results we quote in the article are that when e ≥ 4v, we get a 1/64 factor, and when e ≥ 7.5v, we get a 1/33.75 factor. —David Eppstein (talk) 17:23, 4 July 2008 (UTC)

Ok, I'm fine with thatHermel (talk) 19:01, 16 July 2008 (UTC)