Narrow Results

We define the conformal change and elementary deformation in generalized complex geometry. We apply Swann’s twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish conditions for the Courant integrability of the resulting twisted structures. We associate to any appropriate generalized Kähler manifold $(M,G,{𝒥})$ with a Hamiltonian Killing vector field a new generalized Kähler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when $(M,G,{𝒥})$ is toric, with emphasis on the four-dimensional case, and we apply it to deformations of the standard flat Kähler metric on ${ℂ}^n$, the Fubini–Study metric on $ℂP^2$ and the admissible Kähler metrics on Hirzebruch surfaces. As a further application, we recover the K/K (Kähler/Kähler) correspondence, by specializing to ordinary Kähler manifolds.

We propose a twisted D=N=2 superspace formalism. The relation between the twisted super charges including the BRST charge, vector and pseudoscalar super charges and the N=2 spinor super charges is established. We claim that this relation is essentially related with the Dirac–Kähler fermion mechanism. We show that a fermionic bilinear form of twisted N=2 chiral and anti-chiral superfields is equivalent to the quantized version of BF theory with the Landau type gauge fixing while a bosonic bilinear form leads to the N=2 Wess–Zumino action. We then construct a Yang–Mills action described by the twisted N=2 chiral and vector superfields, and show that the action is equivalent to the twisted version of the D=N=2 super Yang–Mills action, previously obtained from the quantized generalized topological Yang–Mills action with instanton gauge fixing.

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids , where δ_p = σ_p-1σ_p-2 · σ₂σ₁, or equally of the q-string braids . As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

The normal holonomy of a smooth knot in Euclidean space is a geometric invariant with value in the unit circle which is closely related to the writhing number. Here we prove that normal holonomy fibers the space of smooth knots, except for possible singularities on the set of round knots. We deduce results on the existence of isotopies of constant holonomy, writhe and twist.

Detailed analyses and proofs of various formulae used for the calculation and estimation of the writhe of a space curve are given. A new formula for calculating the rate of change of writhe under smooth deformation is presented. This latter result is used to show that the writhe of a closed curve evolving under certain nonlinear evolution equations is conserved.

The large deflection theory of circular cross-section elastic rods is used to consider the writhing of long straight rods subjected to tension and torque, such as undersea cables, and to closed loops with inserted twist, such as DNA supercoils.

The writhed shape of the long straight rod under tension and torque is easily generated by twisting a piece of string with the fingers and consists of three separate parts: a balanced-ply region, a free end loop, and two tail regions. The solution for the rod shape in each of the regions is found. The results are then joined together to ensure continuity of the position and tangent vectors of the strand centreline through the introduction of point forces and moments at the points where the strands enter and exit the balanced ply. The results of the model are consistent with simple experiments on long braided rope.

The writhed shape of the closed loop with twist inserted between the ends prior to closure is modelled as a balanced ply joined to two end loops. The analysis combines the mechanics solution with the conservation of topological link to provide a simple formula which quantitatively predicts the approximate shape and helix angle of the supercoil. The results are in good agreement with simple experiments on rope and with available data on DNA supercoils.