Prime theta-curves on minimal genus Seifert surfaces

TL;DR

This work addresses the classification and construction of prime theta-curves by examining primes knots together with neatly embedded essential arcs on their minimal genus Seifert surfaces. The authors prove that for a prime knot $K$ with a minimal genus Seifert surface $\Sigma$, the union $K\cup\alpha$ is a prime theta-curve for any essential arc $\alpha$ on $\Sigma$, using a geometric analysis with splitting spheres and genus considerations to rule out $\#_2$ and $\#_3$ decompositions. The result yields an infinite, countable family of prime theta-curves and connects to torus-theta-curves, torus knots, and examples like the figure-eight knot inside a Seifert surface for $t(3,4)$, illustrating the broader landscape of knotted graphs in $S^3$. The work also poses questions about the converse and about the reach of this construction to prime theta-curves arising from other knot types, highlighting the interplay between Seifert surface geometry and knotted graph primeness.

Abstract

We prove that each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.

Figures (5)

• Figure 1.1: The theta-graph.
• Figure 1.2: A neatly embedded nonseparating arc $\alpha$.
• Figure 3.1: The simple closed curve $C$ bounds a disk in $S$ that does not contain $\beta$.
• Figure 4.1: Algorithm to construct a minimal genus Seifert surface for a torus knot.
• Figure 4.2: The figure eight knot $J$ in a minimal genus Seifert surface for $t(3,4)$.

Theorems & Definitions (9)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Corollary 4.1
• proof
• Lemma 4.1
• proof