Abstract
We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andruskiewitsch, N., Graña, M.: From racks to pointed Hopf algebras. Adv. Math. 178(2), 177–243 (2003)
Carter, J.S., Kamada, S., Saito, M.: Geometric interpretations of quandle homology. J. Knot Theory Ramif. 10(3), 345–386 (2001)
Carter, J.S., Elhamdadi, M., Saito, M.: Twisted quandle homology theory and cocycle knot invariants. Algebra Geom. Topol. 2, 95–135 (2002)
Carter, J.S., Jelsovsky, D., Kamada, S., Langford, L., Saito, M.: Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Am. Math. Soc. 355(10), 3947–3989 (2003)
Carter, S., Elhamdadi, M., Graña, M., Saito, M.: Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math. 42(3), 499–541 (2005)
Dupont, J.L.: The dilogarithm as a characteristic class for flat bundles. J. Pure Appl. Algebra 44(1–3), 137–164 (1987)
Dupont, J.L., Zickert, C.: A dilogarithmic formula for the Cheeger-Chern-Simons class. Geom. Topol. 10, 1347–1372 (2006)
Etingof, P., Graña, M.: On rack cohomology. J. Pure Appl. Algebra 177(1), 49–59 (2003)
Fenn, R., Rourke, C., Sanderson, B.: Trunks and classifying spaces. Appl. Categorical Struct. 3(4), 321–356 (1995)
Inoue, A.: Quandle and hyperbolic volume. Topol. Appl. 157(7), 1237–1245 (2010)
Inoue, A.: Knot quandles and infinite cyclic covering spaces. Kodai Math. J. 33(1), 116–122 (2010)
Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1), 37–65 (1982)
Kabaya, Y.: Cyclic branched coverings of knots and quandle homology. Pac. J. Math. 259(2), 315–347 (2012)
Kamada, S.: Quandles with good involutions, their homologies and knot invariants, intelligence of low dimensional topology 2006, pp. 101–108, Ser. Knots Everything, 40, World Sci. Publ., Hackensack, NJ (2007)
Matveev, S.: Distributive groupoids in knot theory (Russian). Math. USSR-Sbornik 47, 73–83 (1982)
Meyerhoff, R.: Density of the Chern-Simons invariant for hyperbolic 3-manifolds, low-dimensional topology and Kleinian groups. Lond. Math. Soc. Lect. Note Ser. 112, 217–239 (1986)
Neumann, W.: Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds, Topology ’90. Columbus, OH (1990)
Neumann, W., Yang, J.: Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96(1), 29–59 (1999)
Neumann, W.: Extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 8, 413–474 (2004)
Niebrzydowski, M., Przytycki, J.H.: Homology operations on homology of quandles. J. Algebra 324(7), 1529–1548 (2010)
Weeks, J.: Computation of Hyperbolic Structures in Knot Theory, Handbook of Knot Theory, pp. 461–480. Elsevier B. V, Amsterdam (2005)
Zickert, C.: The volume and Chern-Simons invariant of a representation. Duke Math. J. 150(3), 489–532 (2009)
Acknowledgments
The authors would like to express their sincere gratitude to Professor Sadayoshi Kojima for encouraging them. The first author was supported in part by JSPS Global COE program “Computationism as a Foundation for the Sciences”. The second author was partially supported by JSPS Research Fellowships for Young Scientists. We thank Christian Zickert for for useful conversations. Finally, we also thank the referees for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Inoue, A., Kabaya, Y. Quandle homology and complex volume. Geom Dedicata 171, 265–292 (2014). https://doi.org/10.1007/s10711-013-9898-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-013-9898-2