Hiroshima Mathematical Journal

$N$-degeneracy in rack homology and link invariants

Mohamed Elhamdadi and Sam Nelson

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Abstract

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

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

Dates
First available in Project Euclid: 30 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1333113010

Digital Object Identifier
doi:10.32917/hmj/1333113010

Mathematical Reviews number (MathSciNet)
MR2952076

Zentralblatt MATH identifier
1310.57022

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

Keywords
Finite racks rack homology enhancements of counting invariants cocycle invariants

Citation

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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