New invariants of links in S3 preserved by 4-moves

Study of equivalence classes of links up to n-moves plays an important role in the theory of invariants based on the skein relation and, in particular, skein modules of 3-manifolds. We consider Nakanishi's 4-move conjecture and its modification to 2-component link (homotopically trivial links) proposed by Kawauchi and extended the results concerning Nakanishi 4-move conjecture and Kawauchi's question for links of 2 components. We study invariants of 4-moves derived as derived groups of permutation representations which generalize fundamental groups of cyclic branch covers of S 3 along link. We analyze their potential strength and show, in particular, that our invariant allow us to obtain results of Dabkowski and Przytycki, and Nakanishi concerning 4-move equivalence classes. Moreover, 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-move to the trivial knot. We also prove that links of 2 components with 11 crossings, and links 6 * a1.a2. a3.a4.a 5.a6 such that ai is a 2-algebraic tangle with no trivial components reduce to either 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 as a conterexample for the Kawauchi's question concerning reducibility of links of 2 components by 4-moves. We plan to investigate 4-move equivalence classes of links for non-alternating links with 12-crossings. One of the results, that would make this feasible would be to extend results concerning 4-move reducibility of links with 2 components to the next infinite class of links described by Conway symbol 8*. We are currently working on obtaining similar type of results for 8* as we have obtained for the class 6*.