We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots. 1

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algeb...

Given a multivariate generating function F (z1, . . . , zd) = ∑ ar1,... ,rdz r1 1 · · · z rd d , we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat sin...