A preorder on the set of links with applications to symmetric unions

Abstract

For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^{\mathrm{orb}}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^{\mathrm{orb}}(L)\to G^{\mathrm{orb}}(L^{\prime })$ fitting into a certain commutative diagram, we define a relation $L⪰L^{\prime }$ and explore the relationships between the two links. Specifically, we prove that if $L⪰L^{\prime }$ and $L$ is a Montesinos link with $r$ rational tangles $(r\ge 3)$, then $L^{\prime }$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L^{\prime }$ satisfying $L⪰L^{\prime }$. In contrast, if $L$ has determinant zero, then $L⪰L^{\prime }$ for every $2$-bridge link $L^{\prime }$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation.

44 pages, 1 figure. Minor revision to the definition of $\pi$-domination