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

TL;DR

We address the problem of determining all trace-free ${\rm SL}(2,\mathbb{C})$-representations of $\pi_{1}(S^{3}-L)$ for arborescent links. The authors develop a representation-theoretic framework for tangles, classifying representations into VR$_0$, VR$_1$, VNR$_\ell$ (and analogous HR/VN types) and provide a recursive glueing scheme that assembles local data into global representations; a key tool is the map $\chi$ from a tangle's representation space to a compact parameter set $\mathcal{U}$ that records boundary traces. They prove that the representation spaces of arborescent tangles decompose in a controlled way under horizontal and vertical composition, enabling an explicit method to find all representations of arborescent links, and they illustrate the method with detailed computations for a class of 3-bridge arborescent links. This yields a practical algorithm for computing trace-free ${\rm SL}(2,\mathbb{C})$-representations and has potential implications for branched-cover representations and related link invariants.

Abstract

Given a link $L\subset S^3$, a representation $π_1(S^3-L)\to{\rm SL}(2,\mathbb{C})$ is {\it trace-free} if it sends each meridian to an element with trace zero. We present a method for completely determining trace-free ${\rm SL}(2,\mathbb{C})$-representations for arborescent links. Concrete computations are done for a class of 3-bridge arborescent links.

Figures (8)

• Figure 1: The simplest four tangles: (a) $[0]$, (b) $[\infty]$, (c) $[1]$, (d) $[-1]$
• Figure 2: (a) $T_{1}+T_{2}$; (b) $T_{1}\ast T_{2}$
• Figure 3: (a) the numerator of $T$; (b) a tangle $T$; (c) the denominator of $T$
• Figure 4: Three arcs forming a crossing
• Figure 5: A tangle diagram with the ends directed outwards

Theorems & Definitions (13)

• Lemma 2.1
• Remark 2.2
• Remark 3.1
• Example 3.2
• Proposition 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof