# zbMATH — the first resource for mathematics

Quandle cocycles from invariant theory. (English) Zbl 1287.55002
For a group $$G$$ and a right $$G$$-module $$M$$, one has the structure of a quandle on $$X=G \times M$$ by defining a binary operation $$(a,g)\triangleleft(b,h) = ((a-b)h + b, h^{-1}gh)$$, which is understood as the associated quandle of a structure of a $$G$$-family of quandles on $$M$$, a family of quandle operations on $$M$$ indexed by $$G$$ [A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro, “A $$G$$-family of quandles and handlebody-knots”, arXiv:1205.1855].
Let $$C_{*}^{gr}(M;\mathbb{Z})$$ be the non-homogeneous group chain complex of $$M$$, where $$M$$ is simply regarded as an abelian group. The $$G$$-action on $$M$$ induces an action on $$C_{*}^{gr}(M;\mathbb{Z})$$, and let $$C_{*}^{gr}(M;\mathbb{Z})_{G}$$ be the $$G$$-coinvariants. The author constructs a chain map from a non-homogenous coordinate version of the rack complex of $$X$$ to $$C_{*}^{gr}(M;\mathbb{Z})_{G}$$. Dually, this provides a way to obtain quandle cocycles from $$G$$-invariant group cocycles, and in particular, yields a cocycle of a $$G$$-family of quandles. Since group (co)invariant (co)cycles have been much studied and are easier to treat than quandle cocycles, the author succeeds in constructing various non trivial quandle cocycles explicitly.

##### MSC:
 55N35 Other homology theories in algebraic topology 20J06 Cohomology of groups 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Keywords:
group homology; quandle; invariant theory; knots
Full Text:
##### References:
 [1] Brown, K. S., (Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, (1994), Springer-Verlag New York) [2] Campbell, H. E.A.; Shank, R. J.; Wehlau, D. L., Vector invariants for the two dimensional modular representation of a cyclic group of prime order, Adv. Math., 225, 1069-1094, (2010) · Zbl 1198.13009 [3] Campbell, H. E.A.; Wehlau, D. L., (Modular Invariant Theory, Encyclopaedia of Mathematical Sciences, vol. 139, (2011), Springer-Verlag Berlin) [4] Carter, J. S.; Jelsovsky, D.; Kamada, S.; Langford, L.; Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc., 355, 3947-3989, (2003) · Zbl 1028.57003 [5] Carter, J. S.; Kamada, S.; Saito, M., Geometric interpretations of quandle homology, J. Knot Theory Ramifications, 10, 345-386, (2001) · Zbl 1002.57019 [6] Fenn, R.; Rourke, C.; Sanderson, B., The rack space, Trans. Amer. Math. Soc., 359, 701-740, (2007) · Zbl 1123.55006 [7] Inoue, A.; Kabaya, Y., Quandle homology and complex volume · Zbl 1300.57012 [8] Ishii, A.; Iwakiri, M., Quandle cocycle invariants for spatial graphs and knotted handlebodies, Canad. J. Math., 64, 102-122, (2012) · Zbl 1245.57014 [9] Ishii, A.; Iwakiri, M.; Jang, Y.; Oshiro, K., A $$G$$-family of quandles and handlebody-knots · Zbl 1306.57011 [10] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23, 37-65, (1982) · Zbl 0474.57003 [11] Matveev, S., Distributive groupoids in knot theory, Mat. Sb. (N.S.), 119, 161, 78-88, (1982) [12] Mochizuki, T., The 3-cocycles of the Alexander quandles $$\mathbb{F}_q [T] /(T - \omega)$$, Algebr. Geom. Topol., 5, 183-205, (2005) · Zbl 1085.55004 [13] Neusel, M. D.; Smith, L., (Invariant Theory of Finite Groups, Math. Surveys Monographs, vol. 94, (2002), Amer. Math. Soc) · Zbl 0999.13002 [14] Nosaka, T., On quandle homology groups of Alexander quandles of prime order, Trans. Amer. Math. Soc., 365, 3413-3436, (2013) · Zbl 1336.20050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.