On meridian-traceless \(\mathrm{SU}(2)\)-representations of link groups (Q2692564)

One approach to study knots in \(S^3\) is to study representations of the knot group, i.e.\@ the fundamental group of the knot complement. \textit{P. Kronheimer} and \textit{T. Mrowka} [J. Differ. Geom. 84, No. 2, 301--364 (2010; Zbl 1208.57008)] showed that the knot group of a non-trivial knot admits an irreducible representation in \(SU(2)\) such that the meridian of the knot is mapped to a traceless element in \(SU(2)\). Such an irreducible representation is called \textit{meridian-traceless}. The authors of the paper under review generalize the above result of Kronheimer-Mrowka to links, namely suppose \(L \subset S^3\) is a link. Then it is proven that \(\pi_1(S^3 - L)\) admits an irreducible meridian-traceless representation in \(SU(2)\) if and only if \(L\) is not the unknot, the Hopf link, or a connected sum of Hopf links. A consequence is that \(\pi_1(S^3-L)\) admits an irreducible representation in \(SU(2)\) if and only if \(L\) is neither the unknot nor the Hopf link. The proof uses singular instanton Floer homology theory introduced by \textit{P. B. Kronheimer} and \textit{T. S. Mrowka} [Publ. Math., Inst. Hautes Étud. Sci. 113, 97--208 (2011; Zbl 1241.57017); J. Topol. 4, No. 4, 835--918 (2011; Zbl 1302.57064)]. It is proven that if \(\pi_1(S^3 - L)\) does not admit any irreducible meridian-traceless \(SU(2)\)-representations, then the singular instanton Floer homology group of \(L\) has the minimal possible rank.