A congruence between link polynomials

Abstract

Throughout, ‘link’ means ‘oriented link in S 3 ’, and ‘polynomial’ means ‘Laurent polynomial’. Let L be a link; then denotes the oriented polynomial of L (see [2, 8]), written in Morton's variables (see [5]), and denotes the semi-oriented polynomial of L (see [3]). Let L have components K 1 , …, K n . The total linking of L , denoted t(L) , is the sum of the linking numbers lk ( K i , K j ) with 1 ≤ i < j ≤ n . Let f be a framing of L (that is, f assigns an integer to each K i ); then f(L) denotes the total framing of L (that is, the sum of these integers), and A ( L, f ) denotes the naturally associated union of annuli embedded in S 3 Define the framed polynomial { L, f } ( v, z ) to be (–1) n plus ( v −1 – v ) z −1 times the sum, over all non-empty sublinks K of L , of (where K has k components with 1 ≤ k ≤ n ).