Zentner, Raphael (2017) A class of knots with simple SU(2)-representations. SELECTA MATHEMATICA-NEW SERIES, 23 (3). pp. 2219-2242. ISSN 1022-1824, 1420-9020
Full text not available from this repository. (Request a copy)Abstract
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-(sic)(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-(sic)(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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | HOLOMORPHIC DISKS; FLOER HOMOLOGY; REPRESENTATIONS; INVARIANTS; SU(2); SURGERY; SPACES; LINK; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 14 Dec 2018 13:16 |
| Last Modified: | 21 Mar 2019 16:25 |
| URI: | https://pred.uni-regensburg.de/id/eprint/1570 |
Actions (login required)
 |
View Item |
