Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A resistance invariant of special alternating links

Michal Jablonowski

TL;DR

The paper introduces a new invariant $FP(D)=\operatorname{tr}(L^T L^+)$ for special, reduced, alternating knot and link diagrams, where $L$ is the Laplacian of the associated Tait graph with edge weights $\omega\in\{-1, +1\}$. Despite variations in the Laplacian spectrum under flype moves, $FP(D)$ is invariant across flype-related diagrams, with a key interpretation as the total effective resistance of the corresponding electrical network: $FP(D)=\frac{\omega}{2}\sum_e r(e)$. The authors prove this invariance using Schur complements and the electrical-network perspective, provide explicit computations (e.g., $FP(8a2A)=FP(8a2B)=\frac{8}{3}$), and tabulate $FP(K)$ for several prime special alternating knots. This links graph-theoretic Laplacians to knot-theoretic invariants and offers a concrete, diagrammatic tool for studying special alternating links.

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.

A resistance invariant of special alternating links

TL;DR

The paper introduces a new invariant for special, reduced, alternating knot and link diagrams, where is the Laplacian of the associated Tait graph with edge weights . Despite variations in the Laplacian spectrum under flype moves, is invariant across flype-related diagrams, with a key interpretation as the total effective resistance of the corresponding electrical network: . The authors prove this invariance using Schur complements and the electrical-network perspective, provide explicit computations (e.g., ), and tabulate for several prime special alternating knots. This links graph-theoretic Laplacians to knot-theoretic invariants and offers a concrete, diagrammatic tool for studying special alternating links.

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.
Paper Structure (4 sections, 1 theorem, 13 equations, 6 figures)

This paper contains 4 sections, 1 theorem, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Examples
  4. Proof

Key Result

Theorem 2.2

For special, reduced, alternating diagrams $D_1$, $D_2$ of the same oriented knot or link $KL$, we have $FP(D_1)=FP(D_2)$.

Figures (6)

  • Figure 1: A nugatory crossing.
  • Figure 2: The convention for positive and negative crossings.
  • Figure 3: The diagram $8a2A$ .
  • Figure 4: The diagram $8a2B$ .
  • Figure 5: A flype-move on the tangle $R$ moves the crossing $c$ from one side of $R$ to the other, while rotating $R$ by $180^\circ$ around the horizontal axis.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • proof : Proof of Theorem \ref{['thm:main']}