Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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PARTIALLY ABELIAN REPRESENTATIONS OF KNOT GROUPS

  • Cho, Yunhi (Department of Mathematics University of Seoul) ;
  • Yoon, Seokbeom (Department of Mathematical Sciences Seoul National University)
  • Received : 2016.12.14
  • Accepted : 2016.12.26
  • Published : 2018.01.31
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Abstract

A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called w-variables. In this paper, we consider the case when pinched octahedra appear as a boundary parabolic solution in this decomposition. The w-solution with pinched octahedra induces a solution for a new knot obtained by changing the crossing or inserting a tangle at the pinched place. We discuss this phenomenon with corresponding holonomy representations and give some examples including ones obtained from connected sum.

Keywords

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Figure 1. The 5-term triangulation and region variables.

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Figure 2. Wirtinger generators around a crossing ck.

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Figure 3. A crossing-change and Wirtinger generators.

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Figure 4. The 85 knot diagram and R-related diagrams.

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Figure 5. The 818 knot diagram.

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Figure 6. Rational tangles [3] and [2,？2, 3].

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Figure 7. R-related diagrams: the granny knot, 821, and 815.

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Figure 8. R-related diagrams: the granny knot, 819, and 85.

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Figure 9. R-related diagrams: the square knot, 820, and 810.

References

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  2. J. Cho, Optimistic limit of the colored Jones polynomial and the existence of a solution, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1803-1814.
  3. J. Cho and J. Murakami, Optimistic limits of the colored Jones polynomials, J. Korean Math. Soc. 50 (2013), no. 3, 641-693. https://doi.org/10.4134/JKMS.2013.50.3.641
  4. H. Kim, S. Kim, and S. Yoon, Octahedral developing of knot complement I: pseudo-hyperbolic structure, arXiv:1612.02928.
  5. K. Teruaki and M. Suzuki, A partial order in the knot table, Experimental Mathematics 14.4 (2005), 385-390. https://doi.org/10.1080/10586458.2005.10128937
  6. D. Thurston, Hyperbolic volume and the Jones polynomial, Handwritten note (Grenoble summer school, 1999), 21 pp.
  7. W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton, 1977.
  8. Y. Yokota, On the potential functions for the hyperbolic structures of a knot complement, Invariants of knots and 3-manifolds (Kyoto, 2001), 303-311, Geom. Topol. Monogr., 4, Geom. Topol. Publ., Coventry, 2002.