Linear skein theory and link polynomials
It is well known that the Jones polynomial of an oriented link is fairly stable under the reversal of the orientation of a component. In fact, let \(V_{L_ 1}(t)\) and \(V_{L_ 2}(t)\) be the Jones polynomials of two oriented links \(L_ 1\) and \(L_ 2\), one of which, say \(L_ 2\), is obtained from \(L_ 1\) by reversing the orientation of some of its components. Then \(V_{L_ 1}(t)=t^{\lambda}V_{L_ 1}(t)\) from some integer. In this paper, this result is generalized to the generalized Jones polynomials from a slightly different point of view. Let \(P_ L(1,m)\) denote the polynomial defined by Lickorish and Millett, and \(F_ L(a,z)\) the polynomial defined by L. Kauffman. Let L be an oriented link of n components \(K_ 1,...,K_ n\). Choose one component, say \(K_ n\) and construct an oriented satellite link \(K^*_ n\) (with \(K_ n\) as a companion) in a small tubular neighborhood of \(K_ n\). Thus we obtain a new oriented link \(L_ 1=K_ 1\cup...\cup K_{n-1}\cup K^*_ n\). Now by reversing the orientations of all components of \(K^*_ n\), we will obtain another oriented link \(L_ 2\). Then it is proved that \(P_{L_ 1}(l,m)=P_{L_ 2}(l,m)\) and \(F_{L_ 1}(a,z)=F_{L_ 2}(a,z)\) (Theorems 2 and 3). The author also shows that if L is a mutant of L', then \(F_ L(a,z)=F_{L'}(a,z)\). The paper includes the matrix formula to evaluate \(F_ L(a,z)\) for a 2-bridge link L.
scientific article; zbMATH DE number 4040851