ON 4-MOVE EQUIVALENCE CLASSES OF KNOTS AND LINKS OF TWO COMPONENTS

Abstract

We study equivalence classes of knots and links of 2 components modulo 4-move. We show that all knots up to 12 crossings and knots in the family 6* reduce by 4-moves to the trivial knot. We also prove that links of 2 components with 11 crossings, and links 6* a1.a2.a3.a4.a5.a6 such that ai is a 2-algebraic tangle with no trivial components reduce to either the trivial link or to the Hopf link. For alternating links of 2-components with 12 we show that L reduces by 4-moves to either trivial link or to the Hopf link whenever L is different than 9*.2 : .2 : .2 (or its mirror image). We suggest the alternating link 9*.2 : .2 : .2 with 12 crossings as a potential example to answer the Problem 1.1(iii) in negative.