The Invariants of 4-Moves

This paper introduces 4−moves, a class of tangle replacement moves called rational pq − moves. We demonstrate the strength of invariants of 4 −moves and make modification to some long standing 4 −moves problems like Nakanishi’s 4 − moves conjecture[11]. We present the results and examples to show that Nakanishi’s 4 −moves conjecture is true for all knots upto 12 crossings. We suggest that the link 9∗.2 : .2 : .2 as a counter example to answer his conjecture. Mathematics Subject Classifications: 57M99; 55N20D