S.~K.~Lando
Mutant knots and intersection graphs
This is a report about a joint work with S.~Chmutov.
Finite order knot invariants are described in terms of chord
diagrams --- finite collections of chords with disjoint ends in
a circle. The intersection graph of a chord diagram is the
graph whose vertices are in one-to-one correspondence with the
chords of the diagram, and two vertices are connected by an edge
if the corresponding chords intersect. The intersection graph of a
chord diagram carries only a part of the information encoded in
the diagram. The study of weight systems depending on the
intersection graph of a chord diagram rather than on the diagram
itself was initiated in our joint works with S.~Duzhin more than
ten years ago.
Mutant knots are pairs of knots that differ by rotation of
a two-string tangle. We prove that a finite order knot invariant
does not distinguish mutant knots if and only if the corresponding
weight system depends on the intersection graph only. This theorem
produces a variety of corollaries and leads to a number of
interesting conjectures.