An oriented state model for the Jones polynomial and its applications alternating links

Abstract

In this paper, we de. ne a polynomial invariant of regular isotopy, G(L), for oriented knot and link diagrams L. From G(L) by multiplying it by a normalizing factor, we obtain an ambient isotopy invariant, N-L, for oriented knots and links. We compare the polynomial N-L with the original Jones polynomial and with the normalized bracket polynomial. We show that the polynomial NL yields the Jones polynomial and the normalized bracket polynomial. As examples, we give the polynomial G(L) of some knot and link diagrams and compute the polynomial GL for torus links of type (2, n), and applying computer algebra ( MAPLE) techniques, we calculate the polynomial G(L) of torus links of type (2, n). Furthermore we give its applications to alternating links. (c) 2007 Elsevier Inc. All rights reserved.