Links and cubic 3-polytopes

It is well known that a prime link diagram corresponds to a signed plane graph without cut vertices (Kauffman, 1989). In this paper, we present a new relation between prime links and cubic 3-polytopes. Let $S$ be the set of links such that each $L\in S$ has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except $(2,n)$-torus links and $(p,q,r)$-pretzel links, can be obtained from $S$ by using some operation of untwining. Furthermore, we define the generalized cubic 3-polytope chains and then show that any nontrivial link can be obtained from $\mathbb{S}$ by some untwining operations, where $\mathbb{S}$ is the set of links corresponding to generalized cubic 3-polytope chains. These results are used to simplify the computation of the Kauffman brackets of links so that the computing can be done in a unified way for many infinite families of links.