The representation-polynomial divisibility conjecture for epimorphisms of 2-bridge knot groups

About 18 years old · traced to

Let ρ1:G(K(r1))→SL(2,C)\rho_1:G(K(r_1))\to SL(2,\mathbb C) and ρ2:G(K(r2))→SL(2,C)\rho_2:G(K(r_2))\to SL(2,\mathbb C) be canonical parabolic representations, with representation polynomials f1(z)f_1(z) and f2(z)f_2(z), respectively.

Representation-polynomial divisibility conjecture. If f2(z)f_2(z) divides f1(z)f_1(z), then there exists an epimorphism

G(K(r1))→G(K(r2)).G(K(r_1))\to G(K(r_2)).

The conjecture proposes that divisibility of canonical representation polynomials detects an epimorphism between the corresponding 2-bridge knot groups. The source says the converse to the known implication appears quite likely, but supplies no proof or resolution.

References

Primary source

Mikami Hirasawa and Kunio Murasugi, “Evaluations of the twisted Alexander polynomials of 2-bridge knots at 1”, arXiv:0808.3058 (2008).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.