Difference between revisions of "Rotor(2)"
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==Rotor in graph theory.== | ==Rotor in graph theory.== | ||
| − | The | + | The $n$-rotor of a [[Graph|graph]] is the part of the graph that is invariant under the action of the [[Cyclic group|cyclic group]] $Z_n$; [[#References|[a7]]], [[#References|[a8]]]. |
==Rotor in knot theory.== | ==Rotor in knot theory.== | ||
| − | The | + | The $n$-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 $2 \pi / n$. |
| − | If one modifies the rotor part of a link diagram by rotation of the rotor along an axis of symmetry of an | + | 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|Jones–Conway polynomial]] for $n \leq 4$ and the [[Kauffman bracket polynomial|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|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 [[#References|[a1]]]. It is an open problem (as of 2000) whether the Alexander polynomial is preserved under rotation for any | + | 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 $n$, [[#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==== | ====References==== | ||
| − | <table>< | + | <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> |
Latest revision as of 16:57, 1 July 2020
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 |
How to Cite This Entry:
Rotor(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotor(2)&oldid=50190
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