A resistance invariant of special alternating links

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.

💡 Research Summary

The paper introduces a novel integer invariant FP for special, reduced, alternating diagrams of oriented knots and links. Starting from a checkerboard‑shaded diagram D, the authors construct the Tait graph Γ(D) whose vertices correspond to the unshaded regions and whose directed edges correspond to crossings. Each crossing contributes a weight ω∈{±1} to its edge: +1 for a negative crossing and –1 for a positive crossing. In the special alternating setting all crossings have the same sign, so every edge receives the same weight ω.

The Laplacian matrix L of Γ(D) is defined in the usual way (weighted out‑degree on the diagonal, negative weighted adjacency off‑diagonal). Because the graph is balanced (in‑degree equals out‑degree at each vertex) we have L·1=0 and Lᵀ·1=0. The Moore–Penrose pseudoinverse L⁺ of L is then taken, and the invariant is defined as

  FP(D) = tr(Lᵀ L⁺).

The authors show that FP is not a spectral invariant: two flype‑related diagrams of the knot 8ₐ₂ have Laplacians with different characteristic polynomials, yet both yield the same FP value. The key insight is that FP can be interpreted as the total effective resistance of the weighted graph when viewed as an electrical network. Using the well‑known formula for resistance distance

  r(i,j) = (L⁺){ii} + (L⁺){jj} – 2(L⁺)_{ij},

they define the resistance matrix R =