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기하학 수리물리 연구단
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Octahedral developing of knot complement I: Pseudo-hyperbolic structure

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Title
Octahedral developing of knot complement I: Pseudo-hyperbolic structure
Author(s)
Hyuk Kim; Seonhwa Kim; Seokbeam Yoon
Publication Date
2018-12
Journal
GEOMETRIAE DEDICATA, v.197, no.1, pp.123 - 172
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. © 2018 Springer Science+Business Media B.V., part of Springer Natur
URI
https://pr.ibs.re.kr/handle/8788114/5038
ISSN
0046-5755
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
Files in This Item:
10.1007_s10711-018-0323-8.pdfDownload

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