Related papers: A simple formula for the Casson-Walker invariant
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT…
We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, are necessary and sufficient to express any string link Vassiliev invariant of order two.…
We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livshitz Theorem to Anosov actions by higher-rank abelian groups; it…
The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants…
The Theta invariant is the simplest 3-manifold invariant defined with configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the…
We generalize Milnor link invariants to all types of surface-links in $4$--space (possibly with boundary). This is achieved by using the notion of cut-diagram, which is a 2-dimensional generalization of Gauss diagrams, associated to…
For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched…
We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant…
Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was…
Kashaev and Reshetikhin proposed a generalization of the Reshetikhin-Turaev link invariant construction to tangles with a flat connection in a principal G-bundle over the complement of the tangle. The purpose of this paper is to adapt and…
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous…
We compute two-term skein modules of framed oriented links in oriented 3-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the…
We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant and compute the…
Chern-Simons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and…
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a…
We study invariant statistical connections on the space $\mathcal{N}_0^n$ of zero-mean multivariate normal distributions (the multivariate centered Gaussian model) equipped with the Fisher metric $g^F$. We introduce moduli spaces of…
Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend…
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly…
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot…
We study series invariants for plumbed 3-manifolds and knot complements twisted by a root lattice. Our series recover recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri and apply more generally to 3-manifolds which are…