From: Danny Calegari
Newsgroups: sci.math
Subject: Re: Fields Medals and Nevanlinna Prize
Date: Fri, 21 Aug 1998 13:45:08 -0700
C. Hillman quoted:
> . . . Another result of
> Kontsevich relates to knot theory.
> . . . A key question in knot theory
> is, which of the various knots are equivalent? Or in other words,
> which knots can be twisted and turned to produce another knot without
> the use of scissors? This question was raised at the beginning of the
> 20th century, but it is still unanswered. It is not even clear which
> knots can be undone, that is converted to a simple loop.
This is totally untrue. The homeomorphism problem for knot complements,
and in fact for all "Haken 3-manifolds" was solved by Haken and
Hemion over 20 years ago. The correspondence knot <-> knot complement
was proved more recently by Gordon and Luecke, but is irrelevant for
the question of identifying knots.
The question of which knots can be undone is even easier, and
Hass et. al. have even shown that an unknot projection with n
crossings can be "undone" (i.e. moved to the standard round circle)
in under k(10^6)^n Reidemeister moves, for some explicit constant
k. Note that some experts feel that 10^6 could be replaced by 3,
but that 3 is probably the best possible bound obtainable from
refinements of Haken's approach.
The issue of whether Kontsevich's knot invariants are the "best" or
not is also debatable, but in any case, they are very closely
related to the Vassiliev invariants, which from the point of view
of distinguishing knots, are enormously easier to compute. but the
beauty and interest of Kontsevich's invariants is that they have
deep structural relationships with perturbative Chern-Simons theory
and the (co)homology of Moduli spaces of Riemann surfaces.
BTW, "Kontsevich" is an anagram of "knot chives".
--
" "
- Buster Keaton