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Simple physics ideas are used to derive an exact expression for a flat connection on the complement of a torus knot. The result is of some mathematical importance in the context of constructing representations of the knot group -- a…
Electron transport in two-dimensional conducting materials such as graphene, with dominant electron-electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the…
The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and…
We show how macroscopic manifestations of $P$ (and $T$) symmetry breaking can arise in a simple system subject to Aharonov-Bohm interactions. Specifically, we study the conductivity of a gas of charged particles moving through a dilute…
There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of vector fields…
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by…
Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in…
We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding…
The well known Liouville-Arnold theorem says that if a level surface of integrals of an integrable system is compact and connected, then it is a torus. However, in some important examples of integrable systems the topology of a level…
Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many--particle configuration space. Electron-magnetic field and electron-electron interactions…
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…
In this article, we consider piecewise smooth differential equations $Z_{X_-X_+}$, where $X_-$ and $X_+$ are linear vector fields in dimension 3, having the torus as discontinuity manifold. We consider that $Z_{X_-X_+}$ is an inelastic…
In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\{\Phi_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that…
We study the loop representation of the quantum theory for 2+1 dimensional general relativity on a manifold, $M = {\cal T}^2 \times {\cal R}$, where ${\cal T}^2$ is the torus, and compare it with the connection representation for this…
A nodal loop is formed by band crossing along a one-dimensional closed manifold, with each point on the loop a linear nodal point in the transverse dimensions and can be classified as type-I or type-II depending on the band dispersion.…
We derive the Schr\"odinger equation for a particle confined to the surface of a normal and a binormal helical nanoribbon, obtain the quantum potentials induced by their respective curved surface geometries, and study the localized states…
The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counter-rotating cylinders is considered. A numerical…
The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of…
Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$,…
We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and…