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Quandle homology and complex volume. (English) Zbl 1300.57012

The authors show that the complex volume gives rise to a second cohomology class [cvol] of the quandle \(\mathcal{P}\) that consists of the parabolic elements of \(\mathrm{PSL}(2;\mathbb{C})\). They prove that, for a hyperbolic link \(L\) in \(S^{3}\), by evaluating [cvol] by the quandle homology class \([C(\mathcal{S})]\) that comes from a shadow coloring corresponding to the discrete faithful representation \(\pi_{1}(S^{3}-L) \rightarrow \mathrm{PSL}(2;\mathbb{C})\), one obtains the complex volume of the link complement.
The key technical ingredient is a new variant of quandle homology, which the authors call the simplical quandle homology. This homology is based on simplicial sets, whereas the usual quandle homology is based on cubical sets. They show that there is a homomorphism from the usual quandle homology to the simplicial quandle homology. As the authors explain, in the setting of link complements, this map can be seen as a process of constructing an (ideal) triangulation from a shadow coloring of a link diagram.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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