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==Rotor in graph theory.==
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301702.png" />-rotor of a [[Graph|graph]] is the part of the graph that is invariant under the action of the [[Cyclic group|cyclic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301703.png" />; [[#References|[a7]]], [[#References|[a8]]].

==Rotor in knot theory.==
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301705.png" />-rotor of a link diagram (cf. [[Knot and link diagrams|Knot and link diagrams]]) is the part of the link diagram that is invariant under rotation by an angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301706.png" />.

If one modifies the rotor part of a link diagram by rotation of the rotor along an axis of symmetry of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301707.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301708.png" />, the [[Jones–Conway polynomial|Jones–Conway polynomial]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r1301709.png" /> and the [[Kauffman bracket polynomial|Kauffman bracket polynomial]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017010.png" />. Also, the problem for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017012.png" /> a link and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017013.png" />-rotant share the same space of Fox <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017014.png" />-colourings (cf. [[Fox-n-colouring|Fox <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017015.png" />-colouring]]) has been solved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017016.png" /> not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017017.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017018.png" />.

Rotors can be thought of as generalizing the notion of mutation [[#References|[a1]]]. It is an open problem (as of 2000) whether the Alexander polynomial is preserved under rotation for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130170/r13017019.png" />, [[#References|[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|Statistical mechanics, mathematical problems in]]), where a tangle plays the role of spectral parameter in the [[Yang–Baxter equation|Yang–Baxter equation]], [[#References|[a4]]], [[#References|[a2]]], [[#References|[a5]]], [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Anstee,   J.H. Przytycki,   D. Rolfsen,   "Knot polynomials and generalized mutation"  ''Topol. Appl.'' , '''32'''  (1989)  pp. 237–249</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Hoste,   J.H. Przytycki,   "Tangle surgeries which preserve Jones-type polynomials"  ''Internat. J. Math.'' , '''8'''  (1997)  pp. 1015–1027</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.T. Jin,   D. Rolfsen,   "Some remarks on rotors in link theory"  ''Canad. Math. Bull.'' , '''34'''  (1991)  pp. 480–484</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.F.R. Jones,   "Commuting transfer matrices and link polynomials"  ''Internat. J. Math.'' , '''3'''  (1992)  pp. 205–212</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Traczyk,   "A note on rotant links"  ''J. Knot Th. Ramifications'' , '''8''' :  3  (1999)  pp. 397–403</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W.T. Tutte,   "Codichromatic graphs"  ''J. Combin. Th. B'' , '''16'''  (1974)  pp. 168–174</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.T. Tutte,   "Rotors in graph theory"  ''Ann. Discr. Math.'' , '''6'''  (1980)  pp. 343–347</TD></TR></table>