ON THE QUASITORIC BRAID INDEX OF A LINK

Abstract

Abstract. We de ne new link invariants which are called the quasitoricbraid index and the cyclic length of a link and show that the quasitoricbraid index of link with k components is the product of k and the cy-cle length of link. Also, we give bounds of Gordian distance between the(p;q)-torus knot and the closure of a braid of two speci c quasitoric braidswhich are called an alternating quasitoric braid and a blockwise alternat-ing quasitoric braid. We give a method of modi cation which makes aquasitoric presentation from its braid presentation for a knot with braidindex 3. By using a quasitoric presentation of 10 139 and 10 124 , we canprove that u(10 139 ) = 4 and d (10 124 ;K(3;13)) = 8. 1. IntroductionA link is a disjoint union of circles which is embedded in R 3 and a knot isa link with one component. There are various ways to describe a link, one ofwhich is to present it via a braid. Given a braid, there is a link (the closure ofthe braid) corresponding to the braid. Alexander theorem say that every linkis ambient isotopic to the closure of a braid. Toric braid is a braid which canbe drawn on the standard torus. In 2002, Manturov [8] introduced the notionof a quasitoric braid which is a generalization of toric braid, and proved thatevery link can be presented as the closure of a quasitoric braid. By the virtueof the Manturov’s theorem, one can de ne the quasitoric braid index of linkLas the minimum number of strands of a quasitoric braid which presents thegiven link L.In this paper, we will give a formula to calculate the quasitoric braid in-dex and nd a relationship between the quasitoric braid index and other linkinvariants, like the signature of a link.2. Braids and quasitoric braidsFirstly, we recall the braid index and its properties, see [9] for details. Con-sider the point A