arXiv subjects
Publications and source records attributed to William W Menasco.
The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.
Choose any oriented link type X and closed braid representatives X[+], X[-] of X, where X[-] has minimal braid index among all closed braid representatives of X. The main result of this paper is a `Markov theorem without stabilization'. It asserts that there is a complexity function and a finite set of `templates' such that (possibly after initial complexity-reducing modifications in the choice of X[+] and X[-]which replace them with closed braids X[+]', X[-]') there is a sequence of closed braid representatives X[+]' = X^1->X^2->...->X^r = X[-]' such that each passage X^i->X^i+1 is strictly complexity reducing and non-increasing on braid index. The templates which define the passages X^i->X^i+1 include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index m>= 4 a finite set T(m) of new ones. The number of templates in T(m) is a non-decreasing function of m. We give examples of members of T(m), m>= 4, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.
A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 of Birman and Wrinkle [op cit] establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a Corollary that iterated torus knots are transversally simple.