On fibered commensurability

This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability class contains a unique minimal element, whereas the class of Seifert manifolds fibering over the circle consists of a single commensurability class with infinitely many minimal elements. The situation for non-geometric manifolds is more complicated, and we illustrate a range of phenomena that can occur in this context.Comment: 26 pages, 16 figures; version 2 incorporates referee's comment

Reduction of two-loop Feynman integrals in parametric representation with syzygy trick

Reduction of high-loop Feynman integrals is one of the main tasks in scatting amplitude. In this paper, a new representation of Feynman integrals proposed by Chen in [1,2] is considered. We combined Chen's method with "syzygy" trick to simplify the IBP relations, and successfully canceled the dimensional shift and the unwanted doubled propagators. Moreover, we improved the method to deal with tensor's structure. We demonstrated our method using three two-loop integrals to show our method, and presented the analytical reduction coefficients in the top-sector.Comment: 34 page