Twisted Alexander Polynomials of Hyperbolic Links

Abstract

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.2010 Mathematics Subject Classification. Primary 57M27, Secondary 57M05, 57M25. 1 2 TAKAYUKI MORIFUJI AND ANH T. TRAN Theorem 1.1 (Theorem 4.12). For the double twist link L as in Figure 2, the twisted Alexander polynomial ∆ L,ρ0 (t) associated to ρ 0 determines the Thurston norm. Moreover L is fibered if and only if ∆ L,ρ0 (t) is monic.As is well-known, these topological properties of a 2-bridge link are detected by the reduced Alexander polynomial (see [4], [21], [22]). However there seems to be no a priori reason that the same must be true for the twisted Alexander polynomial.Since a lift of the holonomy representation of a hyperbolic link L is one of the parabolic representations (namely, it is a nonabelian SL(2, C)-representation and the traces of the images of all the meridians of L are two), it is natural to consider the following problem: For an oriented hyperbolic link L and its parabolic representation ρ : π 1 (S 3 \ L) → SL(2, C), does the twisted Alexander polynomial associated to ρ determine the Thurston norm and fiberedness of L? In this paper, we give a partial answer to this question in the case of a 2-bridge link. More precisely we show that not all parabolic representations detect the genus (in this case, the Thurston norm is equivalent to the genus) of a hyperbolic 2-bridge link.This paper is organized as follows. In Section 2, we briefly review some basic materials for the SL(2, C)-character variety of a finitely generated group and the twisted Alexander polynomial of an oriented link associated to a two-dimensional linear representation. In Section 3 we review a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots and state its generalization for oriented hyperbolic links. Section 4 is devoted to the calculation of the loci of the character variety which characterize fiberedness and the genus of a wide family of 2-bridge links. This result can be regarded as a generalization of [12] and [11] which discussed the same problem in the case of knots. In the last section, we give an answer to the question on parabolic representations mentioned above.

Computing twisted Alexander polynomials for Montesinos links

“…Then ρ 75 ε • φ is conjugate to ρ K a for some a and ε ∈ {±1}. However, looking back to (23), we see that ∆ 75,ρ 7 5 ±1 ∆ K,ρ K a . Thus, while the ordinary Alexander polynomial does not detect K 7 5 , TAP does.…”

A survey of the Thurston norm

“…Furthermore, K is fibered if and only if the leading coefficient of T K is equal to 1. Morifuji and Tran [Mori12,MoTr14,MoTr17] showed that Conjecture 9.3 holds for a certain class of 2-bridge knots. Later, Agol and Dunfield [AD20] showed that equality in Conjecture 9.3 holds for all libroid hyperbolic knots in S 3 , including all 2-bridge knots.…”

“…The class of libroid knots is closed under Murasugi sum and contains all special arborescent knots obtained from plumbing oriented bands. See [MoTr17] for a generalization of Conjecture 9.3 for links. 9.4.…”

The ${\rm SL}(2,\mathbb{C})$-character variety of the Borromean link

“…The hyperbolic torsion conjecture proposed in [5] states that for a hyperbolic knot K, the TAP associated to a lift ρ 0 of the holonomy representation detects the genus and fibredness of K. It was generalized to hyperbolic links in [9]. According to [9] Remark 3.4, the conjecture for a m-component alternating link L is just the same as deg ∆ ρ0 L = 4g(L) + 2(m − 2). It is known that the Borromean link is fibred with genus 1.…”