Trace-free \(SL(2,\mathbb{C})\)-representations of arborescent links (Q2322538)

The paper extends results about trace-free representations given in \textit{H. Chen} [J. Knot Theory Ramifications 27, No. 8, Article ID 1850050, 10 p. (2018; Zbl 1401.57008)] to the larger class of arborescent links. Given a link \(L\in S^3\), a representation \(\pi_1(S^3-L) \rightarrow SL(2,\mathbb{C})\) is trace-free if it sends each meridian to an element with trace zero. Trace free representations are of interest since they carry information on the geometry of link complements, for example each trace-free \(SL(2,\mathbb{C})\)-representation of a link \(L\) gives rise to a representation \(\pi_1(M_2(L)) \rightarrow SL(2,\mathbb{C})\), where \(M_2(L)\) is the double covering of \(S^3\) branched along \(L\). The paper presents a method for completely determining trace-free \(SL(2,\mathbb{C})\)-representations for arborescent links. The method uses trace-free representations of arborescent tangles \(T\) (these are two-string tangles that can contain circular components of the link) defined as a homomorphism \(\pi_1(B^3-T) \rightarrow SL(2,\mathbb{C})\) sending each meridian to an element of \(SL^0(2,\mathbb{C})\). The method relies on the build-up of arborescent tangles, starting with rational tangles via iterated horizontal and vertical composition of tangles. In the last section concrete computations are done for a class of 3-bridge arborescent links.