WOVEN BRAIDS AND THEIR CLOSURES

Abstract

A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence tame links can be presented as plats and closures of woven braids. Restricting on knots we get the 'woven version' of the well-known theorem of Markov, giving moves that are capable of producing all woven braids with equivalent closures. As corollary we obtain that a link in which each component is dyed with at least two different colors can be projected on a plane without crossing strands of the same color. The lowest order part of the HOMFLY polynomial of a closed woven braid can be read off directly; a complete characterization of all occuring terms is given. Finally, a table of all minimal woven braids for the 84 prime knots with at most nine crossings is appended. The average word length and the average number of entries per knot type tune out to be suprising small.