[Paper Review] Knot Group Epimorphisms, II

Daniel S. Silver, Wilbur Whitten|ArXiv.org|Jun 19, 2008

Geometric and Algebraic Topology28 references5 citations

TL;DR

This paper investigates epimorphisms between knot groups and their preservation of peripheral structures, focusing on torus and 2-bridge knots. It establishes that for a torus knot $k$, any epimorphism $\pi k \to \pi k'$ implies $k'$ is also a torus knot, and classifies all such $k'$. For 2-bridge knots, it proves that if $k >_p k'$, then $k'$ is a 2-bridge knot with determinant properly dividing that of $k$, and only finitely many such $k'$ exist, resolving a problem posed by J. Simon.

ABSTRACT

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $πk o πk'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we determine the knots $k'$ that can occur. If $k$ is a 2-bridge knot and $k \ge_p k'$, then $k'$ is a 2-bridge knot with determinant a proper divisor of the determinant of $k$; only finitely many knots $k'$ are possible.

Motivation & Objective

• To classify all knots $k'$ for which there exists a peripheral structure-preserving epimorphism $\pi k \to \pi k'$ when $k$ is a torus knot.
• To determine the conditions under which a 2-bridge knot $k$ admits a peripheral epimorphism to another knot $k'$, particularly focusing on determinant and finiteness.
• To address J. Simon’s open problem on the finiteness of such epimorphisms for 2-bridge knots.
• To define and study minimality and $p$-minimality in knot groups, especially for twist knots and torus knots.
• To explore the inheritance of topological invariants (e.g., genus, crossing number, Gromov invariant) under epimorphisms of knot groups.

Proposed method

• The authors define two partial orders on knots: $k \geq k'$ if there exists a knot group epimorphism $\pi k \to \pi k'$, and $k \geq_p k'$ if the epimorphism preserves peripheral structure.
• They use algebraic topology tools, including the centralizer of meridians in the commutator subgroup, to show that meridian-preserving epimorphisms between prime knots preserve peripheral structure.
• For 2-bridge knots, they apply properties of Alexander polynomials and determinant divisibility to constrain possible targets $k'$ of epimorphisms.
• They prove that if $k$ is a $(p_1,q_1)$-2-bridge knot and $k >_p k'$, then $k'$ is a $(p_2,q_2)$-2-bridge knot with $p_2$ properly dividing $p_1$, leading to finiteness.
• They define minimality and $p$-minimality, showing that twist knots are $p$-minimal and that a torus knot is minimal iff both indices are prime.
• They use surgery descriptions and known examples (e.g., Riley’s knot, square knot) to construct explicit epimorphisms that kill the longitude, illustrating zero-degree maps between knot exteriors.

Experimental results

Research questions

• RQ1 For a given torus knot $k$, which knots $k'$ admit a peripheral epimorphism $\pi k \to \pi k'$?
• RQ2 Given a 2-bridge knot $k$, how many knots $k'$ can admit a peripheral epimorphism $\pi k \to \pi k'$, and what constraints do their invariants (e.g., determinant) satisfy?
• RQ3 Does the existence of a meridian-preserving epimorphism $\pi k \to \pi k'$ imply that $k'$ is a 2-bridge knot, and is the number of such $k'$ finite?
• RQ4 Under what conditions is a knot $k$ $p$-minimal, i.e., no nontrivial $k'$ satisfies $k >_p k'$?
• RQ5 Which topological invariants (e.g., genus, crossing number, Gromov invariant) are preserved or inherited under peripheral epimorphisms of knot groups?

Key findings

• If $k$ is a torus knot, then any epimorphism $\pi k \to \pi k'$ preserving peripheral structure implies that $k'$ is also a torus knot, and all such $k'$ are completely classified by the paper.
• For a 2-bridge knot $k$, any $k'$ such that $k >_p k'$ must be a 2-bridge knot with determinant properly dividing that of $k$, and only finitely many such $k'$ exist.
• The paper resolves a problem of J. Simon by proving that for any 2-bridge knot $k$, only finitely many knots $k'$ admit a meridian-preserving epimorphism $\pi k \to \pi k'$.
• Every nontrivial twist knot is $p$-minimal, meaning no nontrivial $k'$ satisfies $k >_p k'$, and for each $n \geq 3$, there exists a $p$-minimal knot with crossing number $n$.
• A $(p_1,p_2)$-torus knot is minimal if and only if both $p_1$ and $p_2$ are prime, establishing a precise algebraic criterion for minimality in this class.
• The genus of $k'$ is at most that of $k$ when $k$ is a 2-bridge or fibered knot, since $\Delta_{k'}(t)$ divides $\Delta_k(t)$ and genus equals half the degree of the Alexander polynomial.