Title: Title: The 3-move conjecture for 5-braids

Author: Qi Chen (Master Degree Thesis)

Preprint: GWUM-1999-10 

Abstract:

Nakanishi conjectured in 1981 that any link is 3-equivalent to a trivial link.

J. Przytycki proved that some classes of links, including 11 crossing links, are 3-equivalent to trivial links. Based on this result we prove that all closed 4-braids are 3-equivalent to trivial links. Then we show that all links with no more than 12 crossings are 3-equivalent to trivial links. For closed 5-braids only 7 of them (up to conjugacy class) have more than 12 crossings. We reduce five of them by hand. The remaining two are 3-equivalent. Hence there is only one unsettled case which is a 5-braid
with 20 crossings. We prove also similar results for t_3, \bar t_4-moves.