A resistance invariant of special alternating links

TL;DR

This paper introduces a new numerical invariant for special alternating links based on the Laplacian matrix of their Tait graphs, which remains unchanged under flype moves and relates to electrical network resistance.

Contribution

The paper defines a novel resistance-based invariant for special alternating links that is invariant under flype moves, linking knot theory with graph Laplacian spectra.

Findings

Invariant remains constant under flype moves

Explicit computations show invariance despite spectral differences

Values computed for several prime alternating knots

Abstract

We introduce a new numerical invariant for special, reduced, alternating diagrams of oriented knots and links, defined in terms of the Laplacian matrix of the associated Tait graph. For a special alternating diagram, the Laplacian encodes both the combinatorics of the checkerboard graph and the crossing signs. While its spectrum depends on the chosen diagram, we show that a specific quadratic trace expression involving the Laplacian and its Moore-Penrose pseudoinverse is invariant under flype moves. The invariant admits an interpretation in terms of total effective resistance of the associated weighted graph viewed as an electrical network. Explicit computations for pairs of flype-related diagrams demonstrate that, although the Laplacian characteristic polynomials differ, the invariant FP coincides. Values for several prime alternating knots are provided.