A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poisson-commute and induce an (S1×R)-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo & Vu Ngoc [PVG1], [PVG2] by means of five invariants.
One of these five invariants is the so-called twisting indexwhich compares the 'distinguished' torus action given near each focus-focus singular fiber to the global toric 'background action'. Although the abstract definition of the twisting index is not very difficult, it has never been abstractly studied or calculated for explicit examples. One of the reasons may have been that there were no known explicit examples of semitoric systems having more than one focus-focus singularity.
In this talk, we
- give various geometric interpretations of the twisting index;
- present a family of compact semitoric systems having two focus-focus points, cf. [HP].
This is a joint project with Joseph Palmer (Rutgers University). We hope that one of the twisting index reformulations will allow us in the future to compute the twisting index for this family of systems.
[HP] Hohloch, S.; Palmer, J.: A family of compact semitoric systems with two focus-focus singularities. arXiv:1710.05746, 27p.
[PVG1] Pelayo, A.; Vu Ngoc, S.: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math. 177 (3), 571-597, 2009.
[PVG2] Pelayo, A.; Vu Ngoc, S.: Constructing integrable systems of semitoric type. Acta Math. 206 (1), 93-125, 2011.