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Simplicial volume of links from link diagrams

Published online by Cambridge University Press:  06 November 2017

OLIVER DASBACH
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, U.S.A. e-mail: [email protected]
ANASTASIIA TSVIETKOVA
Affiliation:
Department of Mathematics and Computer Science, 360 Dr. Martin Luther King Jr. Blvd., Hill Hall 325, Newark, NJ 07102, U.S.A e-mail: [email protected]

Abstract

The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[Ada83] Adams, C. C. Hyperbolic Structures on Link Complements. Ph.D. thesis. University of Wisconsin (1983).Google Scholar
[Ada85] Adams, C. C. Thrice-punctured spheres in hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 287 (1985), no. 2, 645656.Google Scholar
[Ada86] Adams, C. C. Augmented alternating link complements are hyperbolic. Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112 (Cambridge University Press, Cambridge), 1986, pp. 115130.Google Scholar
[Ada15] Adams, C. C. Bipyramids and bounds on volumes of hyperbolic links, preprint (2015).Google Scholar
[CFK+11] Champanerkar, A., Futer, D., Kofman, I., Neumann, W. and Purcell, J. S. Volume bounds for generalised twisted torus links. Math. Res. Lett. 18 (2011), no. 6, 10971120.Google Scholar
[DL06] Dasbach, O. T. and Lin, X.-S. On the head and the tail of the colored Jones polynomial. Comp. Math. 142 (2006), no. 05, 13321342.Google Scholar
[DL07] Dasbach, O. T. and Lin, X.–S. A volumish theorem for the Jones polynomial of alternating knots. Pacific J. Math. 231 (2007), no. 2, 279291.Google Scholar
[DT15] Dasbach, O. and Tsvietkova, A. A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial. Math. Res. Lett. 22 (2015), no. 4, 10471060.Google Scholar
[FKP08] Futer, D., Kalfagianni, E. and Purcell, J. S. Dehn filling, volume and the Jones polynomial. J. Differential Geom. 78 (2008), no. 03, 429464.Google Scholar
[FKP13] Futer, D., Kalfagianni, E. and Purcell, J. S. Guts of surfaces and the colored Jones polynomial. Lecture Notes in Math., vol. 2069 (Springer, Heidelberg, 2013).Google Scholar
[Gro82] Gromov, M. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 599.Google Scholar
[Lac04] Lackenby, M. The volume of hyperbolic alternating link complements. With an appendix by Ian Agol and Dylan Thurston. Proc. London Math. Soc. (3) 88 (2004), no. 1, 204224.Google Scholar
[Pur07] Purcell, J. S. Volumes of highly twisted knots and links. Algebr. Geom. Topol. (2007), no. 7, 93108.Google Scholar
[SW95] Sakuma, M. and Weeks, J. R. Examples of canonical decomposition of hyperbolic link complements. Japan. J. Math. (N. S.) 21 (1995), no. 2, 393439.Google Scholar
[Thu02] Thurston, W. P. The Geometry and Topology of Three-Manifolds, electronic ed. (2002).Google Scholar
[Tsv14] Tsvietkova, A. Exact volume of hyperbolic 2-bridge links. Comm. Anal. Geom. 22 (2014), no. 5, 881896.Google Scholar
[TT14] Thistlethwaite, M. and Tsvietkova, A. An alternative approach to hyperbolic structures on link complements. Algebr. Geom. Topol. 14 (2014), no. 3, 13071337.Google Scholar