Abstract
In the preceding chapter, we explained \(\mathrm{Col}_X(D)\) as a link invariant. This chapter further equips it with gradings in several ways. The original one is introduced by Fenn-Rourke-Sanderson [FRS1, FRS2], and is graded by a homotopy group. After that, from the homological viewpoints, Carter-Jelsovsky-Kamada-Langford-Saito [CJKLS] used quandle cocycles to introduce computable link-invariants (which are called the cocycle invariants). Furthermore, the invariants are generalized to a shadow version, a non-abelian one, and a bigraded one (see Sects. 4.2–4.4 respectively). In this chapter, we study the invariants with various versions in turn. We assume basic knowledge of CW-complexes (see the textbook [Hat]).
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