1 research outputs found
Monte Carlo Computation of the Vassiliev knot invariant of degree 2 in the integral representation
In mathematics there is a wide class of knot invariants that may be expressed
in the form of multiple line integrals computed along the trajectory C
describing the spatial conformation of the knot. In this work it is addressed
the problem of evaluating invariants of this kind in the case in which the knot
is discrete, i.e. its trajectory is constructed by joining together a set of
segments of constant length. Such discrete knots appear almost everywhere in
numerical simulations of systems containing one dimensional ring-shaped
objects. Examples are polymers, the vortex lines in fluids and superfluids like
helium and other quantum liquids. Formally, the trajectory of a discrete knot
is a piecewise smooth curve characterized by sharp corners at the joints
between contiguous segments. The presence of these corners spoils the
topological invariance of the knot invariants considered here and prevents the
correct evaluation of their values. To solve this problem, a smoothing
procedure is presented, which eliminates the sharp corners and transforms the
original path C into a curve that is everywhere differentiable. The procedure
is quite general and can be applied to any discrete knot defined off or on
lattice. This smoothing algorithm is applied to the computation of the
Vassiliev knot invariant of degree 2 denoted here with the symbol r(C). This is
the simplest knot invariant that admits a definition in terms of multiple line
integrals. For a fast derivation of r(C), it is used a Monte Carlo integration
technique. It is shown that, after the smoothing, the values of r(C) may be
evaluated with an arbitrary precision. Several algorithms for the fast
computation of the Vassiliev knot invariant of degree 2 are provided.Comment: 35 pages, 13 figures, LaTeX + RevTeX 4.1, a mistake in the name of
one of the authors has been correcte