# Rotor(2)

## Rotor in graph theory.

The $n$-rotor of a graph is the part of the graph that is invariant under the action of the cyclic group $Z_n$; [a7], [a8].

## Rotor in knot theory.

The $n$-rotor of a link diagram (cf. Knot and link diagrams) is the part of the link diagram that is invariant under rotation by an angle of $2 \pi / n$.

If one modifies the rotor part of a link diagram by rotation of the rotor along an axis of symmetry of an $n$-gon in which the rotor is placed, one obtains a rotant of the original diagram. A link diagram and its rotant share, in some cases, polynomial invariants of links: the Jones polynomial for $n \leq 5$, the Jones–Conway polynomial for $n \leq 4$ and the Kauffman bracket polynomial for $n \leq 3$. Also, the problem for which $n$ and $p$ a link and its $n$-rotant share the same space of Fox $p$-colourings (cf. Fox $n$-colouring) has been solved for $n$ not divisible by $p$, or $n = p$.

Rotors can be thought of as generalizing the notion of mutation [a1]. It is an open problem (as of 2000) whether the Alexander polynomial is preserved under rotation for any $n$, [a3]. P. Traczyk has announced (in March 2001) the affirmative answer to the problem. There is a relation of rotors with statistical mechanics (cf. also Statistical mechanics, mathematical problems in), where a tangle plays the role of spectral parameter in the Yang–Baxter equation, [a4], [a2], [a5], [a6].

How to Cite This Entry:
Rotor(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotor(2)&oldid=50190
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article