Euler continuants, $N$-spherical functors and periodic semi-orthogonal decompositions

Euler continuants are polynomials giving the universal numerators and denominators of finite continued fractions whose coefficients are independent variables. Remarkably, they admit categorical lifts which are certain complexes of functors obtained from iterated adjoints of a single functor. The totalizations of these complexes can be seen as higher analogs of spherical twists and cotwists. They lead to the concept of $N$-spherical functors which correspond to $N$-periodic semi-orthogonal decompositons (usual spherical functors are obtained for $N=4$). Joint work in progress with T. Dyckerhoff and V. Schechtman.