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].

References

[a1]	R.P. Anstee, J.H. Przytycki, D. Rolfsen, "Knot polynomials and generalized mutation" Topol. Appl. , 32 (1989) pp. 237–249
[a2]	J. Hoste, J.H. Przytycki, "Tangle surgeries which preserve Jones-type polynomials" Internat. J. Math. , 8 (1997) pp. 1015–1027
[a3]	G.T. Jin, D. Rolfsen, "Some remarks on rotors in link theory" Canad. Math. Bull. , 34 (1991) pp. 480–484
[a4]	V.F.R. Jones, "Commuting transfer matrices and link polynomials" Internat. J. Math. , 3 (1992) pp. 205–212
[a5]	J.H. Przytycki, "Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics" , Panoramas of Mathematics , Banach Center Publ. , 34 , Banach Center (1995) pp. 121–148
[a6]	P. Traczyk, "A note on rotant links" J. Knot Th. Ramifications , 8 : 3 (1999) pp. 397–403
[a7]	W.T. Tutte, "Codichromatic graphs" J. Combin. Th. B , 16 (1974) pp. 168–174
[a8]	W.T. Tutte, "Rotors in graph theory" Ann. Discr. Math. , 6 (1980) pp. 343–347