The pretzel knot $P(4, -3, 5)$ is not squeezed

TL;DR

This paper addresses squeezedness for a family of three-strand pretzel knots by linking two slice-torus invariants from distinct knot-homology theories. By comparing the Rasmussen invariant $s$ with the $q_M$-invariant from $\\ ext{Z}_2$-equivariant monopole Floer theory, the authors prove that an infinite class of pretzel knots, including $P(4,-3,5)$, are not squeezed, answering a question of Lewark (2024). The work provides explicit computations of $q_M$ for $P(p,q,r)$, leverages Nemethi’s graded-root theory to identify $L$-space conditions on double branched covers, and analyzes two subfamilies (EVEN Y and EVEN X) to establish non-squeezedness through mismatches between $q_M$ and $s/2$. A key outcome is the demonstration that $q_M$ can detect non-squeezedness in cases where $s/2$ alone would be inconclusive, highlighting the nuanced relationship among slice-torus invariants and knot concordance. The results advance tools for detecting non-squeezedness and sharpen understanding of how different homology theories constrain geometric properties of knots.

Abstract

We prove that an infinite family of three-strand pretzel knots is not squeezed. In particular, we show that $P(4, -3, 5)$ is not squeezed. This answers a question posed by Lewark (2024). Our proof is obtained by comparing the Rasmussen invariant with the $q_M$-invariant introduced by Iida and Taniguchi.

Figures (2)

• Figure 1: A surgery diagram (top) and the corresponding plumbing graph (bottom) of $\Sigma_2(P(p,q,r))$, where $p \ge 2$, $q \le -2$, $r \ge 2$, and $1/p + 1/q + 1/r > 0$.
• Figure 2: A surgery diagram (top) and the corresponding plumbing graph (bottom) of $\Sigma_2(-P(p,q,r))$, where $p \ge 2$, $q \le -2$, $r \ge 2$, and $1/p + 1/q + 1/r < 0$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2: see Section \ref{['sec:qM']}
• Definition 2.1
• Remark 2.2
• Theorem 2.3: Feller--Lewark--Lobb Feller-Lewark-Lobb2024Squeezed*Lemma 3.5
• proof
• Theorem 3.1: Iida--Taniguchi Iida-Taniguchi2024Monopoles*Theorem 4.5
• Proposition 4.1: Jabuka Jabuka2010Rational*Theorem 1.18
• Remark 4.2
• Corollary 4.3