English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A quandle equipped with a good involution is referred to as symmetric. It isknown that the cohomology of symmetric quandles gives rise to strong cocycleinvariants for classical and surface links, even when they are not necessarilyoriented. In this paper, we introduce the category of symmetric quandle modulesand prove that these modules completely determine the Beck modules in thecategory of symmetric quandles. Consequently, this establishes suitablecoefficient objects for constructing appropriate (co)homology theories. Wedevelop an extension theory of modules over symmetric quandles and propose ageneralized (co)homology theory for symmetric quandles with coefficients in ahomogeneous Beck module, which also recovers the symmetric quandle (co)homologydeveloped by Kamada and Oshiro [Trans. Amer. Math. Soc. (2010)]. Ourconstructions also apply to symmetric racks. We conclude by establishing anexplicit isomorphism between the second cohomology of a symmetric quandle andthe first cohomology of its associated group.

Generalized (co)homology of symmetric quandles over homogeneous Beck modules