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Mutant knots and intersection graphs. (English) Zbl 1158.57013
This paper describes the Vassiliev invariants not distinguishing mutants. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, a horizontal axis, or an axis perpendicular to the page. Many known knot invariants cannot distinguish mutant knots. Some Vassiliev knot invariants distinguish certain pairs of mutants. The value of a Vassiliev invariant of order at most \(n\) on a singular knot with \(n\) double points depends on the chord diagram of the knot, that is, the source circle \(S^1\) with chords whose ends are the preimages of the double points. Thus any Vassiliev invariant of order at most \(n\) determines a function on chord diagrams with \(n\) chords.
Functions on chord diagrams satisfying the so called four-term relations are called weight systems and correspond to Vassiliev invariant of framed knots. Given a chord diagram, the associated intersection graph is the graph whose vertices correspond to the chords of the diagram, and two vertices are connected by an edge if and only if the corresponding chords intersect. Thus, any function on graphs determines a function on chord diagrams. However, weight systems exist that do not depend on the intersection graph only. The main result of this paper establishes an equivalence between Vassiliev invariants nondistinguishing mutants and unframed weight systems depending on the intersection graphs of chord diagrams only. The same equivalence is true for Vassiliev invariants of framed knots.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
Software:
Knot Atlas
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