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Octahedral developing of knot complement I: Pseudo-hyperbolic structure

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Abstract

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also of cusp shape. We demonstrate explicit solutions for T(2, N) torus knots, J(N, M) knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspects of the knot diagram.

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Notes

  1. We use the term “segment” instead of “edge” for knot diagrams to avoid the confusion with an edge of a triangulation, see Sect. 3.

  2. If one choose a different \(\mathcal {C}:M\rightarrow M_z\), the developing is not unique as a map. But the simplex-wise image of the developing doesn’t change.

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Acknowledgements

We should thank our colleague Insung Park who assisted us in improving the theory. We are also grateful to C. Zickert for email discussions, in particular, about \({\text {PSL }}(2,\mathbb {C})\)-representations and solutions to the gluing equations.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Korea

    Hyuk Kim & Seokbeom Yoon

  2. Center for Geometry and Physics, Institute for Basic Science, Pohang, 37673, Korea

    Seonhwa Kim

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  1. Hyuk Kim
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  2. Seonhwa Kim
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Correspondence to Seonhwa Kim.

Additional information

Hyuk Kim was supported by Basic Science Research Program through the NRF of Korea funded by the Ministry of Education (2015R1D1A1A01057299).

Seonhwa Kim was supported by IBS-R003-D1.

Seokbeom Yoon was supported by Basic Science Research Program through the NRF of Korea funded by the Ministry of Education (2013H1A2A1033354).

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Kim, H., Kim, S. & Yoon, S. Octahedral developing of knot complement I: Pseudo-hyperbolic structure. Geom Dedicata 197, 123–172 (2018). https://doi.org/10.1007/s10711-018-0323-8

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