| Abstract: |
| 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 $(\mathbb S^1 \times \mathbb R)$-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a couple of years ago by Pelayo $\&$ Vu Ngoc by means of five invariants. One of these five invariants is the so-called twisting index which 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 was only very recently calculated for the first time, namely for systems with 1 focus-focus singularity, see Alonso $\&$ Dullin $\&$ Hohloch [arXiv:1712.06402 + 1 preprint in preparation]. In this talk, we present the (results of the) ongoing project with J.\ Alonso (Antwerp) and J.\ Palmer (Rutgers): (a) Calculation of the twisting index for the family of semitoric systems admitting 2 focus-focus singularities in Hohloch $\&$ Palmer [arXiv:1710.05746]. (b) Putting the twisting index in relation with well-known notions from classical dynamical systems like rotation number, winding number, intersection number etc. (c) Observing the change of the twisting index under changes of semitoric systems. |
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