Representation spaces of pretzel knots. (English) Zbl 1234.57018

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 \(R(K,i)/SU(2)\).


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI arXiv


[1] M Boileau, H Zieschang, Nombre de ponts et générateurs méridiens des entrelacs de Montesinos, Comment. Math. Helv. 60 (1985) 270 · Zbl 0569.57001
[2] O Collin, N Saveliev, Equivariant Casson invariants via gauge theory, J. Reine Angew. Math. 541 (2001) 143 · Zbl 0989.57013
[3] R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. \((3)\) 61 (1990) 109 · Zbl 0705.57009
[4] K Fukaya, Floer homology of connected sum of homology \(3\)-spheres, Topology 35 (1996) 89 · Zbl 0848.58010
[5] M Heusener, E Klassen, Deformations of dihedral representations, Proc. Amer. Math. Soc. 125 (1997) 3039 · Zbl 0883.57001
[6] M Heusener, J Kroll, Deforming abelian \(\mathrm{SU}(2)\)-representations of knot groups, Comment. Math. Helv. 73 (1998) 480 · Zbl 0910.57004
[7] A Kawauchi, A survey of knot theory, Birkhäuser Verlag (1996) · Zbl 0861.57001
[8] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 · Zbl 0960.57005
[9] E P Klassen, Representations of knot groups in \(\mathrm{SU}(2)\), Trans. Amer. Math. Soc. 326 (1991) 795 · Zbl 0743.57003
[10] P Kronheimer, T Mrowka, Knot homology groups from instantons · Zbl 1302.57064
[11] P Kronheimer, T Mrowka, Knots, sutures, and excision, J. Differential Geom. 84 (2010) 301 · Zbl 1208.57008
[12] P B Kronheimer, T S Mrowka, Khovanov homology is an unknot-detector, Publ. Math. Inst. Hautes Études Sci. (2011) 97 · Zbl 1241.57017
[13] S Lewallen, Khovanov homology of alternating links and \(\mathrm{SU}(2)\) representations of the link group
[14] W B R Lickorish, An introduction to knot theory, Graduate Texts in Math. 175, Springer (1997) · Zbl 0886.57001
[15] X S Lin, A knot invariant via representation spaces, J. Differential Geom. 35 (1992) 337 · Zbl 0774.57007
[16] N Saveliev, Floer homology of Brieskorn homology spheres, J. Differential Geom. 53 (1999) 15 · Zbl 1025.57033
[17] N Saveliev, Representation spaces of Seifert fibered homology spheres, Topology Appl. 126 (2002) 49 · Zbl 1024.57018
[18] A Shumakovitch, private communication (2010)
[19] A Shumakovitch, Torsion of the Khovanov homology for \(\Z_2\)-thin knots, Lecture notes, Knots in Poland III (Bedlewo, 2010): Conference on Knot Theory and its Ramifications (2010)
[20] A Weil, On discrete subgroups of Lie groups (II), Ann. of Math. \((2)\) 75 (1962) 578
[21] A Weil, Remarks on the cohomology of groups, Ann. of Math. \((2)\) 80 (1964) 149 · Zbl 0192.12802
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.