arXiv research

New examples show non-trivial parity-biquandle bracket.

Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define …

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D>1$. Hence, in contrast with the $D=1$ case, the deformation theory in the multivariable case is non-trivial.

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents,…

A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…

This paper will be an exposition of the Kauffman bracket polynomial model of the Jones polynomial, tangle methods for computing the Jones polynomial, and the use of these methods to produce non-trivial links that cannot be detected by the Jones polynomial.

New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.

This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the pol…

We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Mor…

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinator…

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complic…

Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.

Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Pois…

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result a…

Unified approach to homological representations of topological groups.

New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.

New bracket theory connects three nonholonomic dynamics models.

Flat connections derived from Poisson brackets on loop spaces.

It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…

The paper constructs compatible Poisson brackets on gl(N).

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp…

Researchers attempt to categorify biquandle brackets using Khovanov homology methods.

Explains biquandle brackets and quivers for a topology talk.

A new framework describes dissipation using a metriplectic 4-bracket.

New geometric definition of Lie bracket for undirected curves.

Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.

The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.

In this note we prove that, for a vector bundle $E$ over a manifold $M$, a Dorfman bracket on $TM\oplus E^*$ anchored by $\operatorname{pr}_{TM}$ and with $E$ a vector bundle over $M$, is equivalent to a lift from $Γ(TM\oplus E^*)$ to linear sections of $TE\oplus T^*E\to E$, that intertwines the given Dorfman bracket w…

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

Introduces a new bracket for multicontact geometry and applies it to field theories.

A new invariant for knotted graphs defined by label bracket.

We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of …

Computes Kauffman bracket polynomial for specific 2-tangle shadows.

New Poisson bracket connects to logarithmic manifolds.

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure an…

Paper derives explicit formulas for AJ-bracket of tied links.

Goldman bracket distinguishes surface homeomorphisms.

The paper defines and proves equivalence of nonholonomic brackets in contact mechanical systems.

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathem…

We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive…

A new double quasi-Poisson bracket on surface groups.

Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.