A characterization of quasipositive two-bridge knots

TL;DR

The paper establishes a precise criterion: a two-bridge knot $K(p,q)$ is quasipositive if and only if the negative continued fraction of $p/q$ is even, linking braid-theoretic quasipositivity with number-theoretic data. It connects this criterion to contact topology by showing that the associated lens space $L(p,q)$ supports a tight, Stein-fillable structure with $c_1=0$ exactly in the even-case, and uses this to deduce that smoothly slice two-bridge knots are non-quasipositive, with an independent knot-theoretic proof also provided. It also reinterprets Tanaka’s criterion in terms of regular continued fractions and shows the equivalence of positivity, quasipositivity, and strong quasipositivity for two-bridge links. An Appendix by Orevkov supplements the main text with a Seifert-graph approach that clarifies how to extract key arguments from related work and reinforces the non-quasipositive conclusion for slice two-bridge knots.

Abstract

We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in the Appendix.

Figures (3)

• Figure 1: Two-bridge link $K(p,q)$, where $p/q = [c_1,c_2,\ldots,c_{2m+1}].$
• Figure 2: Alternating diagram $D$ for the rational link $L$.
• Figure 3: The tangle $T_i$ for an odd $i$.

Theorems & Definitions (22)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3: Tanaka t
• Corollary 1.4
• Remark 1.5
• Proposition 1.6
• Proposition 1.7
• proof : Proof of Proposition \ref{['prop: chern']}
• proof : Proof of Theorem \ref{['thm: main']}
• proof : Proof of Proposition \ref{['prop: tan']}