Browse > Article
http://dx.doi.org/10.4134/BKMS.b160996

PARTIALLY ABELIAN REPRESENTATIONS OF KNOT GROUPS  

Cho, Yunhi (Department of Mathematics University of Seoul)
Yoon, Seokbeom (Department of Mathematical Sciences Seoul National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.1, 2018 , pp. 239-250 More about this Journal
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
knot diagram change; boundary parabolic representation;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 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.
2 J. Cho and J. Murakami, Optimistic limits of the colored Jones polynomials, J. Korean Math. Soc. 50 (2013), no. 3, 641-693.   DOI
3 H. Kim, S. Kim, and S. Yoon, Octahedral developing of knot complement I: pseudo-hyperbolic structure, arXiv:1612.02928.
4 K. Teruaki and M. Suzuki, A partial order in the knot table, Experimental Mathematics 14.4 (2005), 385-390.   DOI
5 D. Thurston, Hyperbolic volume and the Jones polynomial, Handwritten note (Grenoble summer school, 1999), 21 pp.
6 W. P. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton, 1977.
7 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.
8 J. Cho, Connected sum of representations of knot groups, J. Knot Theory Ramifications 24.03 (2015), 1550020.   DOI