Rafał M. Siejakowski

Instituto de Matemática e Estatística
Universidade de São Paulo


My research focuses primarily on topological invariants of knots, links and 3-manifolds. In particular, I am interested in various aspects of the Volume Conjecture. Some generalizations of the Volume Conjecture include severeal hypothetical relationships between geometric invariants, including the hyperbolic volume, Reidemeister torsions of 3-manifolds, and the leading and sub-leading terms in asymptotic expansions of certain quantum invariants, such as the partition functions of quantum Chern-Simons theories.

In recent years, several new invariants of this kind have been constructed by the means of state integrals of Turaev-Viro type on triangulations of 3-manifolds, a notable example being the “meromorphic 3D-index”. I have formulated a very concrete kind of “Volume Conjecture” for the meromorphic 3D-index in my joint work with Craig Hodgson and Andrew Kricker.

These conjectural asymptotic approximations of the meromorphic 3D-index led me to discover a new “classical” state integral invariant, whose existence was predicted by Kashaev. This invariant, called the hypergeometric invariant, can be described in terms of state integrals in which the tetrahedral weights are given by the Euler beta function. It is still far from clear what this invariant means topologically and whether it is related to any previously known quantities.


I rendered this visualization of an ideal tetrahedron in using Blender.