The integer values SU(3) Casson invariant for Brieskorn spheres. (English) Zbl 1098.57008
A method for calculating the integer valued SU(3) Casson invariant \(\tau_{\text{SU}(3)}\) [H. U. Boden, C. M. Herald and P. Kirk, Math. Res. Lett. 8, No. 5–6, 589–603 (2001; Zbl 0991.57014)] for the Brieskorn homology spheres is described. Examples are given for a number of cases.
For example, \(\tau_{\text{SU}(3)}(\Sigma(3,5,15n\pm7)) = 276n^2 \pm254n + 56\) and \(\tau_{\text{SU}(3)}(\pm\Sigma(4,27,108n-1)) = 959595n^2 - 19569n\). (\(\Sigma(4,27,108n-1)\) is obtained from \(1/n\) surgery on the torus knot \(K_{4,27}\)).
In all cases in the paper \(\tau_{\text{SU}(3)}\) is even and its growth with \(n\) (as above) is quadratic.
For example, \(\tau_{\text{SU}(3)}(\Sigma(3,5,15n\pm7)) = 276n^2 \pm254n + 56\) and \(\tau_{\text{SU}(3)}(\pm\Sigma(4,27,108n-1)) = 959595n^2 - 19569n\). (\(\Sigma(4,27,108n-1)\) is obtained from \(1/n\) surgery on the torus knot \(K_{4,27}\)).
In all cases in the paper \(\tau_{\text{SU}(3)}\) is even and its growth with \(n\) (as above) is quadratic.
Reviewer: Lee P. Neuwirth (Princeton)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57R57 | Applications of global analysis to structures on manifolds |
