Linear Bounds of the Crosscap Number of Knots

TL;DR

This work generalizes linear bounds on the crosscap number from alternating links to Conway sums of strongly alternating tangles, tying $C(L)$ to Jones-polynomial data via $T_L = |eta_L| + |eta_L'|$ and the twist number of diagrams. It proves two-sided bounds for two-tangle sums and extends these to $l$-tangle sums, highlighting both $C(L)$- and $T_L$-dependent expressions and their relationships to closures $K_{iN}$ and $K_{iD}$. The paper further shows that $T_L$ and $C(L)$ can grow independently by constructing families (torus knots and Whitehead doubles) where one quantity remains bounded while the other grows arbitrarily, establishing limits to universal linear bounds in terms of Jones-polynomial coefficients. These results illuminate the interaction between knot invariants and geometric complexity, and point to open questions about extending bounds to all hyperbolic knots and broader classes of tangles.

Abstract

Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certain coefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for the crosscap number of Conway sums of strongly alternating tangles. Then we find families of links for which these coefficients of the Jones polynomial and the crosscap number grow independently. These families will enable us to show that neither linear bound generalizes for all links.

Figures (12)

• Figure 1: Left: A tangle inside a box with directional strands labelled, Center: numerator closure, Right: denominator closure.
• Figure 2: An example of a Conway sum of $l$ tangles.
• Figure 3: Here we see a tangle contained within a Conway Sphere. The blue dots represent the intersection of $T_i$ with $\Sigma$. Then the dotted lines are the intersections of $S$ with $\Sigma$.
• Figure 4: This figure shows a case where $L$ is meridianally boundary compressible. As $\phi_1 \cup \alpha$ create a meridian of $N(L)$.
• Figure 5: Here we put a crossing into Menasco form, with the over strand running across the top of the ball and the lower strand running along the bottom.

Theorems & Definitions (66)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5