Research ArticlesNo Access

Character varieties of $(- 2, 2 m + 1, 2 n)$ -pretzel links and twisted Whitehead links

Anh T. Tran

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

https://doi.org/10.1142/S0218216516500073Cited by:7 (Source: Crossref)

Abstract

We consider some families of links which include $(- 2, 2 m + 1, 2 n)$ -pretzel links and twisted Whitehead links. We calculate the ${SL}_{2} (ℂ)$ -character varieties of these families, and determine the number of irreducible components of these character varieties.

Keywords:

AMSC: 57M27

References

1. G. Burde, $S U (2)$ -representation spaces for two-bridge knot groups, Math. Ann. 288 (1990) 103–119. Crossref, Web of Science, Google Scholar
2. G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics, Vol. 5 (de Gruyter, Berlin, 2003). Google Scholar
3. M. Culler and P. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117(1) (1983) 109–146. Crossref, Web of Science, Google Scholar
4. R. Fricke and F. Klein, Uber die theorie der automorphen modulgruppen, Kgl. Ges. d. W. Nachrichten Math. Phys. Klasse 4 (1896) 91–93. Google Scholar
5. S. Harada, Canonical components of character varieties of arithmetic two-bridge link complements, preprint (2011), arXiv:1112.3441. Google Scholar
6. E. Landes, Identifying the canonical component for the Whitehead link, Math. Res. Lett. 18(4) (2011) 715–731. Crossref, Web of Science, Google Scholar
7. E. Landes, On the canonical components of character varieties of hyperbolic two-bridge link complements, Ph.D. thesis, The University of Texas at Austin (2011). Google Scholar
8. T. Le and A. Tran, The Kauffman bracket skein module of two-bridge links, Proc. Amer. Math. Soc. 142(3) (2014) 1045–1056. Crossref, Web of Science, Google Scholar
9. A. Lubotzky and A. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 336 (1985) 117 pp. Google Scholar
10. M. Macasieb, K. Petersen and R. van Luijk, On character varieties of two-bridge knot groups, Proc. Lond. Math. Soc. 103 (2011) 473–507. Crossref, Web of Science, Google Scholar
11. F. Nagasato and A. Tran, Some families of minimal elements for a partial ordering on prime knots, preprint (2013), arXiv:1301.0138. Google Scholar
12. K. Petersen and A. Tran, Character varieties of double twist links, preprint (2014), arXiv:1411.0758. Google Scholar
13. R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc. 334(1) (1992) 389–409. Crossref, Web of Science, Google Scholar
14. W. Thurston, Three-dimensional Geometry and Topology, Vol. 1, ed. S. Levy, Princeton Mathematical Series, Vol. 35 (Princeton University Press, Princeton, NJ, 1997). Crossref, Google Scholar
15. A. Tran, The universal character ring of the $(- 2, 2 m + 1, 2 n)$ -pretzel link, Internat. J. Math. 24(8) (2013), Article ID: 1350063, 13 pp. Link, Web of Science, Google Scholar
16. M. Vogt, Sur les invariants fondamentaux des equations differentielles linéaires du second ordre, Ann. Sci. École. Norm. Supér. Troi. (3) 6 (1889) 3–71. Crossref, Google Scholar