From:  (Greg Kuperberg)
Newsgroups:     sci.math.research
Subject:        Re: Mutations and Knots
Date:           Fri, 12 Nov 1993 17:36:10 GMT
Organization:   University of Chicago
Keywords:       Mutations, Knots, Vassiliev-invariants

In article <2c08v0$> (  Oliver Dasbach   ) writes:
>1.) How can I decide whether two knots K and L are mutants of one another?

Well, the conclusive method would be to find all possible ways to repeatedly
mutate K, which is not easy, but I think it is possible by Haken's
algorithms for incompressible surfaces.  By Thurston geometry, a knot
has only finitely many mutant equivalents.   In the main case, when
K is hyperbolic, mutation preserves hyperbolic volume, and there are
only finitely many hyperbolic manifolds with a given volume.

If you consider Haken-type algorithms too laborious, you can in
practice use various invariants to try to conclude that two knots
aren't mutants.  As already mentioned, there is hyperbolic volume.  But
also there is the Dehn invariant, as Danny Ruberman explained to me.
Recall Hilbert's problem of determining when two polyhedra are
equivalent by polyhedral dissection.  Dehn found an invariant beyond
volume which distinguishes, for example, cubes from regular tetrahdra.
As it happens, the Dehn invariant also works in hyperbolic geometry,
and moreover if two polyhedral are fundamental domains for the same
hyperbolic group, they are dissection equivalent.  So hyperbolic
groups have a Dehn invariant too.  Finally, mutation can also
be understood as a polyhedral dissection in the hyperbolic case,
so mutation classes have a Dehn invariant also.

Ruberman also told me that the Dehn invariant is a projection of
the Chern-Simons invariant of a hyperbolic manifold (this is not
the same as the Chern-Simons path integral of Witten).  I do not know
whether Chern-Simons detects mutation.

Besides hyperbolic geometry, you can also consider quantum invariants.
Following Reshetikhin and Turaev and a little bit of Witten, a compact
Lie group G and an irreducible G-module V determine a polynomial knot
invariant.  It is well-known that when G = SL(n), O(n), or SP(2n), the
invariants for all V's come from cablings and twisted cablings of
HOMFLY or Kauffman.  (I am principally excluding representations of
Spin(n) and exceptional Lie groups.)  It was noticed early on that many
of these invariants are mutation invariant.  Here a broad condition on
the pair (G,V) that suffices for mutation invariance of the
corresponding knot polynomial:  V tensor V should be multiplicity-free,
i.e., it should be a direct sum of distinct irreducible summands.  This
condition is almost certainly also necessary, although I do not have a
complete argument in my head for the exceptional groups and spinorial
representations (except I think that I have a complete argument for

>2.) In which case is a knot K the only knot within its mutancy class?

First, to mutate, you need to find a four-point tangle in K which is
not rational.  In some cases, such as all torus knots I think, no such
tangle exists.  Also, in the hyperbolic case, if there is only one knot
of the same volume as K, then mutation on K must yield K.

>3.) Is a Vassiliev-invariant v(i) known such that there exist (non-equivalent)
>    knots K and L, mutants of each other, and v(i)(K)<>v(i)(L)?

Sure.  The derivatives at q=1 of quantum invariants are of Vassiliev type.
Consider a quantum invariant from a pair (G,V) where V tensor V is
not multiplicity free; for example you can take G = sl(n) for n>3
and V to be adjoint.  Then the quantum invariant will detect mutation.

More explicitly, for a knot K, take the bidirectional 2-cabling, i.e.,
replace K by two parallel copies of K, and orient one in direction and
the other in the other direction.  Then HOMFLY of this 2-cabling should
detect mutation in K most of the time.