arXiv research

Universal knot polynomials for 2- and 3-strand torus knots are presented.

Study extends Vogel's universality to torus knots in adjoint representation.

The study connects curve intersections with complex stereographic projections to analyze their topological properties.

Revives Vogel's diagrammatic technique for universal Lie algebra computations.

We construct a finitely-presented group such that its Vogel-Levine localization is not transfinitely nilpotent. This answers a problem of J. P. Levine.

Develops method to construct Lie algebra weight system kernel using Vogel algebra.

Generalizes Kauffman-Vogel polynomials to oriented and unoriented 4-valent graphs.

Derives adjoint polynomials of torus knots in explicit form.

Vogel's construction links knot invariants to Lie algebras, revealing new insights.

Study of isometries on hyperbolic 3-manifold cusps.

The article extends a knot polynomial algorithm for singular knots.

Study of knot polynomials for twist satellites, generalizing cabling.

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman-Vogel graph polynomial as a state sum of the Murakami-Ohtsuki-Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the sl(N) Mur…

In [2] Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, satisfying three skein relations, and is defined in terms of a state-sum an…

New invariants for Legendrian graphs defined and proven.

In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.

The paper computes group factors and properties of Wilson loops in Chern-Simons theory.

The theory of Vassiliev invariants deals with many modules of diagrams on which the algebra Lambda defined by Pierre Vogel acts. By specifying a quadratic simple Lie superalgebra, one obtains a character on Lambda. We show the coherence of these characters by building a map of graded algebras beetwen Lambda and a quoti…

Generalizes Jones polynomial to singular knots.

Kricker constructed a knot invariant Z^{rat} valued in a space of Feynman diagrams with beads. When composed with the so called "hair" map H, it gives the Kontsevich integral of the knot. We introduce a new grading on diagrams with beads and use it to show that a non trivial element constructed from Vogel's zero diviso…

New 4D shapes can't be opened like books.

In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three variables, $A$, $B$ and $a$, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=…

In [AU2] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [TUY] and the abelian analog in [AU1]. We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vog…

A new quantum relation connects exceptional Lie algebras and knots.

A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a …

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimensio…

In a previous paper we constructed classical spin Chern-Simons for any compact Lie group $G$: a gauge theory whose action depends on the spin structure of the 3-manifold. Here we apply geometric quantization to the classical Hamiltonian theory and investigate the formal properties of the partition function in the Lagra…

In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge…

New non-semisimple TQFTs derived from quantum Hennings invariants.

By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated…

Study involutions on spaces to compute nonnegatively curved metrics dimensions.

The graded algebra Lambda defined by Pierre Vogel is of general interest in the theory of finite-type invariants of knots and of 3-manifolds because it acts on the corresponding spaces of connected graphs subject to relations called IHX and AS. We examine a subalgebra Lambda_0 that is generated by certain elements call…

Maximal cubic quotient of braid algebra studied for n ≤ 5.

For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{d…

Aimed at geometric applications, we prove the homology cobordism invariance of the $L^2$-betti numbers and $L^2$-signature defects associated to the class of amenable groups lying in Strebel's class $D(R)$, which includes some interesting infinite/finitenon-torsion-free groups. The proofs include the only prior known c…

Constructs new topological theories in 2D not fitting standard axioms.

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

Study bifurcation in plane-to-plane germs with specific properties.

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

Extends results on automorphism groups of flat Minkowski planes to toroidal circle planes.

Stable planes are locally isomorphic to classical projective planes.

Only vertical planes are asymptotic to other planes in 3D space.

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Criteria for sharksfin and deltoid singularities from plane to plane, with applications.

Crooked planes in 3D Minkowski space can be foliated.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…