Epimorphisms between $2$-bridge knot groups and their crossing numbers

Abstract

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K^{\prime }$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K^{\prime })$. Especially it is shown that $c(K)$ is greater than or equal to $3c(K^{\prime })$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.

13 pages

References in corpus (5)

• Epimorphisms between 2-bridge link groups
• Twisted Alexander polynomials and a partial order on the set of prime knots
• Knot Group Epimorphisms, II
• Non-meridional epimorphisms of knot groups
• Knot Group Epimorphisms

Cited by in corpus (3)

• Classification of parabolic generating pairs of Kleinian groups with two parabolic generators
• Symplectic quandles and parabolic representations of 2-bridge Knots and Links
• Generating function on epimorphisms between $2$-bridge knot groups