Epimorphisms of knot groups onto free products

Simon’s conjecture for two-bridge knots

“…Indeed, using [2,3,10,12], one can say more about the nature of the homomorphisms in Theorem 1.2. In particular, we establish: Corollary 1.3.…”

“…In [12], examples are given where the domain knot is small; for example there is an epimorphism of the group of the knot 8 20 onto the group of the trefoil-knot for which the longitude of 8 20 is mapped trivially.…”

“…Indeed, from [12], the property given by Proposition 1.8 can be viewed as an extension of Property Q * of Simon.…”

“…Let K be a non-trivial knot. Simon's Conjecture (see [16, Although this conjecture dates back to the 1970s, and has received considerable attention recently (see [2,3,12,20,22,23,27,28] to name a few), little by way of general results appears to be known. Conjecture 1.1 is easily seen to hold for torus knots (we give the proof in Section 3.1).…”

Non-meridional epimorphisms of knot groups

“…In particular, they constructed many explicit examples of epimorphisms between knot groups. Gonzaléz-Acũna and Ramínez studied which knot groups (particulary those of 2-bridge knots) admit epimorphisms onto torus knot groups [GAR01,GAR03]. In work of Ohtsuki, Riley, and Sakuma [ORS08], Hoste and Shanahan [HS10], and Lee and Sakuma [LS12], systematic constructions of epimorphisms between 2-bridge knot (and link) groups were presented and studied.…”