The ${\rm SL}(2,\mathbb{C})$-character variety of an arborescent knot

Abstract

We clarify steps for determining the ${\rm SL}(2,\mathbb{C})$-character variety of any arborescent knot. Interestingly, we show that the `excellent parts' of arborescent knots $K_1,K_2$ are isomorphic if $K_1$ can be related to $K_2$ through certain moves on projection diagrams. Furthermore, we give a sufficient condition in terms of diagram for the existence of components of dimension larger than $1$. A generalization to arborescent links is sketched.

Figures (8)

• Figure 1: A representation satisfies $\rho(c)=\rho(a)\rho(b)\rho(a)^{-1}$ whenever the directed arcs $a,b,c$ form a crossing in this way.
• Figure 2: Left: a tangle $T\in\mathcal{T}_2^2$; the four end arcs are denoted by $T^{\rm nw}$, $T^{\rm ne}$, $T^{\rm sw}$, $T^{\rm se}$, all directed outward. Middle: the numerator $N(T)$. Right: the denominator $D(T)$.
• Figure 3: Left: $T_1\ast_vT_2$. Right: $T_1\ast_hT_2$.
• Figure 4: Left: the tangle $[1]$. Right: $[-1]$.
• Figure 5: Constructing the rational tangle $[[k_1],\ldots,[k_s]]$; the case $s$ is odd (resp. even) is shown at left (resp. right).

Theorems & Definitions (29)

• Conjecture 1.1
• Lemma 2.1: c.f. Ch22 Lemma 2.2
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Remark 3.3