Quantum fundamental group of knot and its $SL_2$ representation

Abstract

The theory of bottom tangles is used to construct a quantum fundamental group. On the other hand, the skein module is considered as a quantum analogue of the $SL(2)$ representation of the fundamental group. Here we construct the skein module of a knot complement by using the bottom tangles. We first construct the universal space of quantum representations, which is a quantum analogue of the fundamental group, and then factor it by the skein relation to get the skein module. We also investigate the action of the quantum torus to the boundary of complement, and derive the recurrence relation of the colored Jones polynomial, which is known as $A_q$ polynomial.

Figures (66)

• Figure 1: Plat closure of $b \in B_{2k}$.
• Figure 2: Generators $x_i$ and $y_i$ of $\pi_1(D_k, p)$ and $\pi_1(D_{2k}, p)$.
• Figure 3: The action of the braid generator $\sigma_i$ on $\pi_1(D_{2k})$.
• Figure 4: Pushing a trivial element of $\pi_1(S^3\setminus K, p)$ along the plat presentation of the trefoil knot to get a relation of $\pi_1(S^3\setminus K, p)$.
• Figure 5: Inclusions $\iota_1 : D_k \to D_{2k}$ and $\iota_2 : D_{2k} \to S^3 \setminus K$ where $K = \widehat{b}$.

Theorems & Definitions (61)

• Definition 1
• Definition 2
• Proposition 1.1
• proof
• Definition 3
• Definition 4
• Proposition 1.2
• Proposition 1.3
• proof
• Proposition 1.4