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On a conjecture of Dunfield, Friedl and Jackson

In this short note, we show that the twisted Alexander polynomial associated to a parabolic SL(2,C)-representation detects genus and fibering of the twist knots. As a corollary, a conjecture of Dunfield, Friedl and Jackson is proved for the hyperbolic twist knots.


INTRODUCTION
Let K be a knot in the 3-sphere S 3 and denote its knot group by G(K). That is, G(K) = π 1 E(K) where E(K) is the knot exterior S 3 \int(N(K)) which is a compact 3-manifold with torus boundary. For a nonabelian representation ρ : G(K) → SL(2, C), the twisted Alexander polynomial ∆ K,ρ (t) ∈ C[t ±1 ] is defined up to multiplication by some t i , with i ∈ Z, see [10], [16] and [9] for details.
If K is a hyperbolic knot, namely the interior of E(K) admits the complete hyperbolic metric with finite volume, there is a discrete faithful representation ρ 0 : G(K) → SL(2, C), which is called the holonomy representation, corresponding to the hyperbolic structure.
The hyperbolic torsion polynomial T K (t) ∈ C[t ±1 ] was defined in [3] for hyperbolic knots as a suitable normalization of ∆ K,ρ 0 (t). It is a symmetric polynomial in the sense that T K (t −1 ) = T K (t), which seems to contain geometric information. In fact Dunfield, Friedl and Jackson conjectured in [3] that T K determines the genus g(K) and moreover, the knot K is fibered if and only if T K is monic.
They show in [3] that the conjecture holds for all hyperbolic knots with at most 15 crossings. Our main theorem in this note is the following. Theorem 1.1. For all hyperbolic twist knots K (see Figure 1) T K determines the genus g(K) and moreover, the knot K is fibered if and only if T K is monic.
As far as we know, this is the first infinite family of knots for which the conjecture is verified. Since twist knots are 2-bridge knots (in particular alternating knots), their genus and fibering can be detected by the Alexander polynomial (see [2], [12], [13], [14]). However there seems to be no a priori reason that the same must be true for T K . See [3, Section 7], [8] for twisted Alexander polynomials and character varieties of knot groups.
Recall that an SL(2, C)-representation ρ is called parabolic if the meridian of G(K) is sent to a parabolic element of SL(2, C) and ρ(G(K)) is nonabelian. Since the holonomy representation of hyperbolic knots is parabolic, the above theorem is an immediate consequence of the following: Theorem 1.2. Let K be a twist knot and ρ : G(K) → SL(2, C) a parabolic representation. In the next section, we quickly review twisted Alexander polynomials of twist knots for parabolic SL(2, C)-representations (see [11,Sections 3,4] for details). The proof of Theorem 1.2 will be given in Section 3.
Hence γ q (u) never vanishes for the parabolic representations. Thus deg ∆ K,ρ (t) = 2 and hence it determines the genus.
A similar argument applied to γ q (u) − 1 shows that ∆ K,ρ (t) is not a monic polynomial for the nonfibered twist knot K = J(2, 2q) with |q| > 1. This completes the proof of Theorem 1.2.

Remark 3.2.
For the 3830 nonfibered 2-bridge knots K(a, b) with b < a ≤ 287, Dunfield, Friedl and Jackson numerically compute the twisted Alexander polynomials for the parabolic representations. In fact, it is shown in [3, Section 7.6] that ∆ K,ρ (t) is nonmonic and determines the knot genus in every case. Remark 3.3. Let δ q (u) be the second coefficient of ∆ K,ρ (t) in Lemma 2.3. As we saw in Example 2.4, δ ±1 (u) are integers for the fibered twist knots J(2, ±2). It is not so hard to show that δ q (u) ∈ Z[u] and deg δ q (u) = 2q − 4 for q > 1. Therefore we can conclude that δ q (u) with q > 2 is not a rational number for the parabolic representations. On the other hand, we have ∆ K,ρ (t) = (u 2 + 4) − 4t + (u 2 + 4)t 2 for the hyperbolic twist knot K = J(2, 4). In particular, we see that the second coefficient δ 2 (u) is an integer for the holonomy representation, although J(2, 4) is nonfibered (see [3, Section 6.5]).