Hiroshima Mathematical Journal

$N$-degeneracy in rack homology and link invariants

Mohamed Elhamdadi and Sam Nelson

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The aim of this paper is to define a homology theory for racks with finite rank $N$ and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in S. Nelson, Link invariants from finite racks, arXiv:0808.0029. For this purpose, we prove that $N$-degenerate chains form a sub-complex of the classical complex defining rack homology. If a rack has rack rank $N=1$ then it is a quandle and our homology theory coincides with the CKJLS homology theory. Nontrivial cocycles are used to define invariants of knots and examples of calculations for classical knots with up to $8$ crossings and classical links with up to $7$ crossings are provided.

Article information

Hiroshima Math. J., Volume 42, Number 1 (2012), 127-142.

First available in Project Euclid: 30 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Finite racks rack homology enhancements of counting invariants cocycle invariants


Elhamdadi, Mohamed; Nelson, Sam. $N$-degeneracy in rack homology and link invariants. Hiroshima Math. J. 42 (2012), no. 1, 127--142. doi:10.32917/hmj/1333113010. https://projecteuclid.org/euclid.hmj/1333113010

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  • N. Andruskiewitsch, and M. Graña From racks to pointed Hopf algebras. Adv. Math. 178 (2003), no. 2, 177–243.
  • D. Bar-Natan, Ed. The Knot Atlas, http://katlas.math.toronto.edu/wiki/MainPage
  • S. Carter, M. Elhamdadi, M. Graña and M. Saito. Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math. 42 (2005), no. 3, 499–541.
  • S. Carter, M. Elhamdadi, and M. Saito. Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles. Fund. Math. 184 (2004), 31–51.
  • S. Carter, M. Elhamdadi, and M. Saito. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002), 95–135.
  • S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989.
  • S. Carter, S. Kamada and M. Saito. Surfaces in 4-space. Encyclopaedia of Mathematical Sciences, 142. Low-Dimensional Topology, III. Springer-Verlag, Berlin, 2004.
  • J. Ceniceros and S. Nelson. Virtual Yang-Baxter Cocycle Invariants. Trans. Amer. Math. Soc. 361 (2009), no. 10, 5263–5283.
  • P. Etingof and M. Graña. On rack cohomology. J. Pure Appl. Algebra 177 (2003), no. 1, 49–59.
  • R. Fenn; C. Rourke and B. Sanderson, The rack space. Trans. Amer. Math. Soc. 359, (2007), no. 2, 701–740.
  • R. Fenn; C. Rourke and B. Sanderson, James bundles and applications. Proc. London Math. Soc. 3, 89 (2004), no. 1, 217–240.
  • R. Fenn and C. Rourke. Racks and links in codimension two. J. Knot Theory Ramifications 1 (1992) 343-406.
  • R. Litherland and S. Nelson. The Betti numbers of some finite racks. J. Pure Appl. Algebra 178 (2003), no. 2, 187–202.
  • A. Navas, and S. Nelson. On symplectic quandles. Osaka J. Math. 45 (2008), no. 4, 973–985.
  • {S. Nelson.