Journal of the
Korean Mathematical Society JKMS

We define new link invariants which are called the {\it quasitoric braid index} and the {\it cyclic length} of a link and show that the quasitoric braid index of link with $k$ components is the product of $k$ and the cycle length of link. Also, we give bounds of Gordian distance between the $(p,q)$-torus knot and the closure of a braid of two specific quasitoric braids which are called an alternating quasitoric braid and a blockwise alternating quasitoric braid. We give a method of modification which makes a quasitoric presentation from its braid presentation for a knot with braid index $3$. By using a quasitoric presentation of $10_{139}$ and $10_{124}$, we can prove that $u(10_{139})=4$ and $d^\times(10_{124},K(3,13))=8$.

Keywords: link, knot, braid, toric braid, quasitoric braid, braid index, quasitoric braid index

MSC numbers: Primary 57M25, 57M27