Tangle sums and factorization of A-polynomials

Abstract

We show that there exist infinitely many examples of pairs of knots, K_1 and K_2, that have no epimorphism $π_1(S^3\setminus K_1) \to π_1(S^3\setminus K_2)$ preserving peripheral structure although their A-polynomials have the factorization $A_{K_2}(L,M) \mid A_{K_1}(L,M)$. Our construction accounts for most of the known factorizations of this form for knots with 10 or fewer crossings. In particular, we conclude that while an epimorphism will lead to a factorization of A-polynomials, the converse generally fails.

11 pages, 4 figures