Torus theta-curves

TL;DR

The paper studies prime theta-curves arising from adding an arc to a torus knot on a standard torus in $S^3$, showing that every nontrivial torus knot joined with an essential arc yields a prime theta-curve. It then provides an explicit classification of all torus theta-curves up to ambient isotopy and up to homeomorphism with reflections, proving that prime torus theta-curves are precisely the $ heta(p,q)$ with $\gcd(p,q)=1$ and $|p|,|q|\ge 2$, while Kinoshita's theta-curve is not torus-representable. The constituent knots of $ heta(p,q)$ are given by explicit formulas involving torus knots $t(a,b)$ and modular inverses, enabling special families such as consecutive Fibonacci numbers to yield prime examples. Overall, the results illuminate the structure of knotted theta-curves on tori and provide concrete invariants for classification.

Abstract

A natural approach to construct a prime theta-curve is to an add an arc to a prime knot. We study that approach for knots and arcs on a standard torus in the 3-sphere. We prove that each nontrivial torus knot union an essential arc is a prime theta-curve. By comparing the constituent knots of those theta-curves, we obtain infinitely many prime theta-curves. We then classify all theta-curves that lie on a standard torus. In particular, Kinoshita's theta-curve does not lie on a standard torus.

Figures (21)

• Figure 1.1: Trefoil knot $K$ union an inessential arc (left) and $K$ union an essential arc (right).
• Figure 1.2: Theta-curve $\theta(3,5)$ equal to the torus knot $t(3,5)$ union the arc $e_3$.
• Figure 2.1: Two inessential sccs in a surface $\Sigma$ with one boundary component (left) and an inessential arc in $\Sigma$ (right).
• Figure 2.2: Separating essential scc (left) and nonseparating essential scc (right).
• Figure 2.3: Separating essential arc (left) and nonseparating essential arc (right).

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['lem:3sumnoinv']}
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Lemma \ref{['lem:chopubap']}
• Lemma 2.5
• proof : Proof of Lemma \ref{['pkb']}
• proof : Proof of Lemma \ref{['lem:2sumUPrimeKnot3sum']}