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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

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.

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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.

Acknowledgement

Supported by : NRF of Korea

References

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