Cyclic rotation of braid-word closures #
Cyclically rotating a braid word cuts its closed braid between two different levels. The closure diagram therefore does not change: only its crossing and half-edge names do. This file describes that renaming explicitly, proves equality of the resulting oriented PD-codes, and concludes that the two closures are Reidemeister equivalent.
For a word w and a rotation distance k, List.rotateIndexEquiv w k sends an index in
w.rotate k to the index of the same letter in w. Its inverse renames the old crossings, and
the induced crossing-block equivalence renames their four half-edges. Along every strand position
the rotation carries the crossings of w.rotate k to a cyclic rotation of those of w, so by
TauCeti.BraidWord.closure_eq_relabel_of_isRotated these renamings account for the entire closure
construction, including arcs which cross the cut.
Taking w = u ++ v and k = u.length specializes the theorem to the familiar equality of the
closures of u ++ v and v ++ u. This is the diagram-level content of cyclic conjugation, the
word-level generator needed for the conjugation part of Markov equivalence.
Main results #
TauCeti.BraidWord.closure_rotate: rotating a braid word changes its closure only by the explicit induced renaming.TauCeti.BraidWord.reidemeisterEquiv_closure_rotate: a braid word and its rotation have Reidemeister equivalent closures.
References #
- J. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies 82 (1974), Chapter 2.
- W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Proposition 16.10.
Rotating a braid word changes its oriented closure PD-code only by renaming crossings and
half-edges. The inverse of rotateIndexEquiv sends each old crossing name to its name in the
rotated word, and PDCode.crossingBlockEquiv applies the same renaming to all four crossing
slots.
Rotating a braid word gives a Reidemeister equivalent closure.