Colorings of symmetric unions and partial knots

TL;DR

This work defines symmetric unions and partial knots $J$, and studies how $p$-colorings interact with symmetric union presentations. By introducing coloring matrices and a block-structured transformation, the authors prove the main bound $\text{col}_p(J) \leq \text{col}_p(K) \leq \dfrac{(\text{col}_p(J))^2}{p}$ for a symmetric union knot $K$ with partial knot $J$, and deduce a negative answer to whether equal determinants imply symmetric relatability. They construct for any $m$ a family of $2^m$ knots with identical determinants that are pairwise not symmetrically related, demonstrating the non Converse of Lamm’s relation. The paper also discusses conditions for equality in the bound and raises several open questions about the structure and extent of symmetric relations among knots.

Abstract

Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If both $J$ and $J'$ are partial knots for different symmetric union presentations of the same ribbon knot $K$, the knots $J$ and $J'$ are said to be symmetrically related. Lamm proved that if $J$ and $J'$ are symmetrically related, then $\det J = \det J'$, asking whether the converse is true. In this article, we give a negative answer to Lamm's question, constructing for any natural number $m$ a family of $2^m$ knots with the same determinant but such that no two knots in the family are symmetrically related. This result is a corollary to our main theorem, that if $J$ is the partial knot in a symmetric union presentation for $K$, then $\text{col}_p(J) \leq \text{col}_p(K) \leq \frac{(\text{col}_p(J))^2}{2}$, where $\text{col}_p(\cdot )$ denotes the number of $p$-colorings of a knot.

Figures (7)

• Figure 1: The general form used in the construction of a symmetric union.
• Figure 2: At top, an initial setup to construct a symmetric union with partial knot the trefoil. At bottom left, a symmetric union $D \sqcup -D(1)$ for the knot $K = 6_1$. At bottom right, a symmetric union $D \sqcup -D(3)$ for the knot $K = 9_{46}$.
• Figure 3: A 3-coloring of the partial knot $3_1$ (left) induces a 3-coloring of a symmetric union presentation of $6_1$ (right).
• Figure 4: A symmetrically compatible 3-coloring of $9_{46}$ (left) induces a 3-coloring of $3_1 \# 3_1^*$ (right).
• Figure 5: A 3-coloring of $3_1 \# -\!3_1$ that is not symmetrically compatible.

Theorems & Definitions (17)

• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Remark 1.6
• Proposition 2.1
• Corollary 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• proof