Quantum invariants : a study of knots, 3-manifolds, and their sets

By 大槻知忠. 2002

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Quantum invariants of 3‐manifolds via link surgery presentations and non‐semi‐simple categories (2012, background): “Moreover, R gives rise to a braiding cV,W : V ⊗ W → W ⊗ V on C defined by v ⊗ w → τ(R(v ⊗ w)), where τ is the permutation x ⊗ y → y ⊗ x (see [27, 30]).”
On the AJ conjecture for knots (2011, unknown): “t(ML +M−1L−1). Proposition 1.1. One has (−1)nhS n−1(λ),Θ(ℓ)i = Υ(ℓ)JK(n) for all ℓ ∈ S(T2). ON THE AJ CONJECTURE FOR KNOTS 5 Proof. We know from the properties of the Jones-Wenzl idempotent (see e.g. [Oh]) that hSn−1(λ)·µ, ∅i = (t2n +t−2n)hSn−1(λ),∅i hSn−1(λ)·λ, ∅i = hSn(λ)+Sn−2(λ),∅i hSn−1(λ)·λ1,1,”
Problems on invariants of knots and 3-manifolds (2004, background): “where [n] = (qn/2 − q−n/2)/(q1/2 − q−1/2), and U+ denotes the trivial knot with +1 framing, and Qsl2;Vn(K) denotes the quantum invariant of K associated with the irreducible n-dimensional representation of the quantum group Uq(sl2); for details see [221] (see also [321] for the notation).”
Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions (2002, background): “[42], we see that τPsln(C) does not separate L(64, 9) and L(64, 25).”