Algebraic & Geometric Topology

Twisted quandle homology theory and cocycle knot invariants

J Scott Carter, Mohamed Elhamdadi, and Masahico Saito

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Abstract

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums. The invariants are used to derive information on twisted cohomology groups.

Article information

Source
Algebr. Geom. Topol., Volume 2, Number 1 (2002), 95-135.

Dates
Received: 27 September 2001
Accepted: 8 February 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882686

Digital Object Identifier
doi:10.2140/agt.2002.2.95

Mathematical Reviews number (MathSciNet)
MR1885217

Zentralblatt MATH identifier
0991.57005

Subjects
Primary: 57N27 57N99: None of the above, but in this section
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} 57T99: None of the above, but in this section

Keywords
quandle homology cohomology extensions dihedral quandles Alexander numberings cocycle knot invariants

Citation

Carter, J Scott; Elhamdadi, Mohamed; Saito, Masahico. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002), no. 1, 95--135. doi:10.2140/agt.2002.2.95. https://projecteuclid.org/euclid.agt/1513882686


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