A class of knots with simple SU(2)-representations

• Raphael Zentner ^[1]
  1. [1] University of Regensburg

     University of Regensburg

     Kreisfreie Stadt Regensburg, Alemania

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 3, 2017, págs. 2219-2242
• Idioma: inglés
• DOI: 10.1007/s00029-017-0314-x
• Enlaces
  • Texto completo (pdf)
• Resumen

  • We call a knot in the 3-sphere SU(2)-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in SU(2) are binary dihedral. This is a generalization of being a 2-bridge knot. Pretzel knots with bridge number ≥3 are not SU(2)-simple. We provide an infinite family of knots K with bridge number ≥3 which are SU(2)-simple. One expects the instanton knot Floer homology I (K) of a SU(2)-simple knot to be as small as it can be—of rank equal to the knot determinant det(K). In fact, the complex underlying I (K) is of rank equal to det(K), provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of SU(2)-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we obtain the result that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group. This makes use of a non-vanishing result of Kronheimer–Mrowka.