Abstract
In this chapter we show various generalizations of the volume conjecture. Firstly, we introduce the complexification of the conjecture by studying the imaginary part of \(\log {J_N(K;\exp (2\pi \sqrt {-1}/N))}\). We expect the (\(\mathrm {SL}(2;\mathbb {C})\)) Chern–Simons invariant to appear. Secondly, we refine the conjecture by considering more precise approximation of the colored Jones polynomial. We conjecture that the Reidemeister torsion would appear. Lastly, we perturb \(2\pi \sqrt {-1}\) in \(\exp (2\pi \sqrt {-1}/N)\) slightly and see what happens to the asymptotic expansion of the colored Jones polynomial. The corresponding topological phenomenon is to perturb the hyperbolic structure of the knot complement, provided the knot is hyperbolic. If the knot is non-hyperbolic we expect various representations of the fundamental group of the knot complement to \(\mathrm {SL}(2;\mathbb {C})\).
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Notes
- 1.
The first author learned it from Kashaev. Writing \(\operatorname {Re}{w}=x\) and \(\operatorname {Im}{w}=y\), \(\tan {w}=\frac {\cosh {y}\sin {x}+\sqrt {-1}\sinh {y}\cos {x}}{\cosh {y}\cos {x}-\sqrt {-1}\sinh {y}\sin {x}}\). So \(\tan {w}\to \sqrt {-1}\) when y →∞ and \(\tan {w}\to -\sqrt {-1}\) when y →−∞.
- 2.
The proof in [64] is wrong but the statement remains true, which was informed by Ka Ho Wong.
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Murakami, H., Yokota, Y. (2018). Generalizations of the Volume Conjecture. In: Volume Conjecture for Knots. SpringerBriefs in Mathematical Physics, vol 30. Springer, Singapore. https://doi.org/10.1007/978-981-13-1150-5_6
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