arXiv research

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summa…

First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is Schouten bracket. We have several results about Euler number and Betti numbers of …

In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, aske…

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

We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyp…

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with $g$ boundary points and $n$ crossings in the Kauffman bracket skein module is a linear combination of $O(2^g)$ basis elements, with…

The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bra…

For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link…

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the cla…

S. Nelson, M. Orrison, V. Rivera {\cite{S}} modified Kauffman's construction of bracket. Their invariant $Φ^β_X$ takes value in a finite ring $Z_2[t]/(1+t+t^3)$. In this paper, the author generalizes this invariant. The new invariant takes value in a polynomial ring. Furthermore, for a tricolorable link diagram, the au…

Bounding twist number of surface links using polynomial coefficients.

Extends knotoid theory to multi-linkoids, studying various invariants.

We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well known triviality result exists, the Miura equivalence class…

Computes Jones polynomial for specific knots.

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…

New hierarchies and equations derived from Poisson structures.

The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.

In this paper the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general cas…

This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs pres…

We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws of averages under strong dependence, and exhibit uniform mixing. Our results ar…

We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…

Constructs maps on skein modules using non-semisimple quantum invariants.

We show that if {L_n} is any infinite sequence of links with twist number tau(L_n) and with cyclotomic Jones polynomials of increasing span, then lim sup tau(L_n)=infty. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main t…

New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.

Based on the presentation of the Kauffman bracket skein module of the torus given by the third author in previous work, Charles D. Frohman and Răzvan Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure …

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound on the span of the bracket polynomial of K as 4c_2(K), to the n-crossing number:…

New bracket theory connects three nonholonomic dynamics models.

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,…

Flat connections derived from Poisson brackets on loop spaces.

This paper investigates multi-task reinforcement learning in non-Markovian decision making, showing benefits in sample efficiency.

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 algebra counts components of arborescent knots and links.

Study links in 3-manifolds, linking volume to polynomial coefficients.

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

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms and knots.

Study of knots and links in 2-complexes, defining linking numbers and polynomials.

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…

Quantum approach to volume computation from colored Jones polynomials.

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…

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

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

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the surfaces by certain graphs.