Representation spaces of pretzel knots. (English) Zbl 1234.57018

Zentner, Raphael

Algebr. Geom. Topol. 11, No. 5, 2941-2970 (2011).

In the construction of framed instanton knot Floer homology, there appear at the chain group level representation spaces \(R(K,i)\) of knots in \(SU(2)\) with meridian mapped to traceless matrices. Moreover, it is proved that the Khovanov homology of certain knots is isomorphic to the integer homology of \(R(K,i)\).
In this paper, the author proves that this isomorphism is no longer true for alternating pretzel knots \(P(p,q,r)\) with \(p,q,r\) pairwise coprime and such that \(|p|,|q|,|r|>1\). For this end, he gives a complete picture of the representation spaces \(R(K,i)\) by describing the conjugacy classes of representations of \(P(p,q,r)\) by triangles on the \(2\)-sphere and proving that each conjugacy class is isolated in \(R(K,i)/SU(2)\).

Reviewer: Leila Ben Abdelghani (Monastir)

MSC:

57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

pretzel knot; representation space; presentation of the group of pretzel knots; instanton knot Floer homology

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