Mutation (knot theory)

The Kinoshita – Terasaka knot and the Conway knot are mutated apart.

In knot theory , a branch of topology , mutation is an operation that turns one node into another node in a certain way.

As a rule, a mutant differs from the original node, but both nodes have much in common and, in particular, many node invariants are the same. It is therefore a difficult knot theoretical problem to distinguish knots from their mutants.

The term was introduced by John Horton Conway in the 1960s in connection with the Tangle notation for tabulating knots and interweaving.

definition

Tangle operations Left: A Tangle a and its reflection ^- a . Above right: Tangle addition a + b . Middle right: Tangle product ab , equivalent to ^- a + b . Bottom right: Ramification a, b , equivalent to ^- a + ^- b

One looks at a sphere (“Conway sphere”) which intersects the knot (after suitable deformation) in 4 symmetrical points. The part of the knot lying inside the sphere (“tangle”) is rotated by 180 ° or tilted so that two points are exchanged in pairs, and then connected again with the part of the knot lying outside. The new knot created in this way is called the mutation of the original knot.

The effect of the mutation on the nodal complements can be described as follows. The complement of the original knot is cut open along a quadruple dotted sphere and then glued again by means of a hyperelliptic involution of the quadruple dotted sphere. The result is homeomorphic to the complement of the mutant.

Invariants

Many node invariants are the same for nodes and their mutants: the Alexander polynomial , the Jones polynomial , the HOMFLY polynomial , the colored Jones polynomial . Ruberman proved in 1987 that the mutation of a hyperbolic node is hyperbolic again and has the same hyperbolic volume.

It is therefore difficult to distinguish nodes from their mutants. An invariant that can be used to distinguish some nodes from their mutants is Heegaard-Floer homology . The Kinoshita Terasaka and Conway knots shown on the right differ in their Seifert gender .

Josh Greene proved that for alternate nodes, the following three statements are equivalent:

L emerges from K by mutation
${\ displaystyle \ Sigma (L)}$ is homeomorphic too ${\ displaystyle \ Sigma (K)}$
${\ displaystyle {\ widehat {HF}} (\ Sigma (L)) = {\ widehat {HF}} (\ Sigma (K))}$

Here, the 2-fold superposition of the 3-sphere branched along K denotes and the Heegaard-Floer homology . ${\ displaystyle \ Sigma (K)}$ ${\ displaystyle {\ widehat {HF}}}$

literature

John H. Conway “An enumeration of knots and links and some of their related properties”, 1970.
Daniel Ruberman: "Mutation and volumes of knots in S ³ ", Inventiones Mathematicae 90, no.1, 189-215 (1987).
Stephan M. Wehrli: "Contributions to Khovanov homology", dissertation Zurich (2007).
Alexander Stoimenow, Toshifumi Tanaka, “Mutation and the colored Jones polynomial”, Journal of Gökova Geometry & Topology 3, 44-78 (2009).
Joshua E. Greene, "Lattices, Graphs and Conway Mutation," pdf
Hugh R. Morton, Peter R. Cromwell: Distinguishing mutants by knot polynomials. J. Knot Theory Ramifications 5 (1996) no. 2, 225-238. pdf