Talk:Knot invariant

Untitled

Moved from article:

• More needed. Suggestions:
  • n-colourabilty

This is a special case of quandles. Perhaps the quandles entry should be updated to make this connection clearer. --C S 09:07, Nov 10, 2004 (UTC)

Yes, but n-colourability is easier to understand. --Dylan Thurston 22:05, 8 May 2005 (UTC)

• • the twist (I think that's the wrong term - I mean the one where you add together the crossings including signs - writhe, maybe)

It sounds like you mean "writhe". --C S 09:07, Nov 10, 2004 (UTC)

• • for genus, we obviously need Seifert sufaces, which could really do with some pictures.
  • braid index - probably needs an article on braids
  • unknotting number
  • linking number? Really a link invariant
  • relationships between invariants
  • Alexander polynomial
  • Jones polynomial and generalisations
  • homology of knots?

--C S 09:01, Nov 10, 2004 (UTC)

Some more invariants:

• Vassiliev or finite-type invariants
• knot energies: ropelength, etc.

--Dylan Thurston 22:05, 8 May 2005 (UTC)

Other suggestion:

It is noted in the 2nd paragraph there is no known knot polynomial "even which distinguishes just the unknot from all other knots", so the article should also note that the categorified invariants Heegaard Floer homology and Khovanov homology (also mentioned in the article) do detect the unknot.

The A-polynomial and augmentation polynomial are two examples of polynomial invariants that distinguish the unknot from other knots.

--Stephen Hancock 23:50, Feb 20, 2012 (UTC)

Fary-Milnor theorem

This page says that the Fary-Milnor theorem is if and only if. However, the page for that theorem implies that only one direction holds. I don't know enough about the theorem to be comfortable touching either page, though.

The statement was corrected 2006 August 27. -- Chuck 13:24, 14 June 2007 (UTC)

hyperbolic knots generic

I made a few modifications today. I fixed the statement that claimed that because hyperbolic knots were generic, their volumes were knot invariants. The proper attribution is Mostow-Prasad rigidity. But by doing this I also erased the statement that most knots are hyperbolic -- because this is false unless you're very precise on how you generate your knots. If you're using a PL random walk in R^3 then you almost always get a knot with more than one trefoil summand. Humm, I wonder why I'm not finding a proper reference for this -- there's a pair of papers on this kind of random knotting in a conference proceedings from Japan, sometime near 1980. I've heard people claim that "most knots are hyperbolic" but I haven't seen anyone state how one generates knots in a way to ensure this happens. Anyhoo, enough for now. Rybu (talk) 07:45, 25 October 2008 (UTC)

Maybe (as N goes to infinity) a uniformly random knot among the knots with ≤ N crossings has probability approaching 1 of being hyperbolic? Or maybe one has to restrict to prime knots? —David Eppstein (talk) 14:22, 25 October 2008 (UTC)

Here's my feeling on the two main reasons people often say "most knots are hyperblic". One is that if you look at the knot tables, most of them are hyperbolic. Of course, the knot tables only list prime knots (and count mirror images as the same knot), but the use of "knot" in this way is a well-known sloppiness.

The other is that people really mean "hyperbolic knots are plentiful" and through further sloppiness it becomes "most knots..."

References for the random knotting result are Nick Pippenger and a joint paper by Dewitt Sumners and Whittington.--C S (talk) 16:08, 28 October 2008 (UTC)