Knots, tangles, and electrical networks

Abstract

The signed graphs of tangles or of tunnel links (special links in (R3-two parallel lines)) are two terminal signed networks. The latter contain the two terminal passive electrical networks. The conductance across two terminals of a network is defined, generalizing the classical electrical notion. For a signed graph, the conductance is an ambient isotopy invariant of the corresponding tangle or tunnel link. Series, parallel, and star triangle methods from electrical networks yield techniques for computing conductance, as well as giving the first natural interpretation of the graphical Reidemeister moves. The conductance is sensitive to detecting mirror images and linking. The continued fraction of a rational tangle is a conductance. Algebraic tangles correspond to two terminal series parallel networks. For tangles, the conductance can be computed from a special evaluation of quotients of Conway polynomials and there is a similar evaluation using the original Jones polynomial. © 1993 Academic Press, Inc.