The Jones Polynomial of a Connect Sum is Multiplicative: A New Approach Via Trip Matrices

TL;DR

Problem: whether the Jones polynomial $V_K$ is multiplicative under connect sums. The authors adapt the trip-matrix method to an elementary state-sum framework, using $T_K$ as an $n\times n$ matrix over $\mathbb{Z}_2$ and toggled diagonals $T_{K_S}$ to encode states. They show that $w(K)=\sum w(K_i)$, $nul(T_{K_S})=\sum nul(T_{i,S_i})$, and $A(S)=\sum A(S_i)$, $B(S)=\sum B(S_i)$, so the term-by-term contributions factor. Consequently, $V_K=\prod_i V_{K_i}$, providing an accessible, linear-algebraic route to the classical multiplicativity result. The work offers an elementary alternative to skein-based proofs by exploiting a block-diagonal trip-matrix structure for composite knots.

Abstract

We utilize the trip matrix method of calculating the Jones Polynomial to give an alternative proof that the Jones Polynomial is multiplicative under connect sums.

Figures (3)

• Figure 1: The Figure 8 Knot.
• Figure 2: Different labelings for the figure 8 knot.
• Figure 3: A composite knot with three factors

Theorems & Definitions (22)

• Example 2.1
• Definition 2.2: Row-Column Swap
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Definition 2.5: $\Delta$-equivalent
• Remark 2.6
• Definition 3.1: State
• Theorem 3.2