Rknots (version 1.3.2)

mVA: Compute the multivariable Alexander polynomial of a polygonal link

Description

Computes the multivariable Alexander polynomial (MVA) of a polygonal link.

Usage

mVA(points3D, ends, normalized = FALSE, return.A = FALSE)

Arguments

points3D

an $N$ x 3 matrix of the $x$, $y$, $z$ coordinates of a polygonal link

ends

a vector of positive integers defining the separators of the polygonal link

normalized

logical, if FALSE (default) the multivariable non normalized MVA is returned, the normalized MVA otherwise

return.A

logical, if TRUE the Alexander matrix is returned in a format that can be parsed to sympy

Value

the multivariable Alexander polynomial

Details

The polynomial computation relies on rSymPy. Please notice that the first time sympy is invoked is expected to be much slower than subsequent ones.

References

Alexander J. W. (1928) Topological invariants of knots and links. Trans. Amer. Math. Soc. 30: 275-306. Conway J. H. (1970) An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Proc. Conf.,Oxford, 1967), Pergamon, Oxford: 329-358.

Murakami J. (1993) A state model for the multivariable Alexander polynomial. Pacific J. Math. 157, no. 1: 109-135.

Archibald J. (2008) The weight system of the multivariable Alexander polynomial. Acta Math. Vietnamica. 33: 459-470. Archibald J. (2010) The Multivariable Alexander Polynomial on Tangles. PhD Thesis, Department of Mathematics University of Toronto

Torres G. (1953) On the Alexander polynomial Ann. Math. 57: 57-89.

Comoglio F. and Rinaldi M. (2011) A Topological Framework for the Computation of the HOMFLY Polynomial and Its Application to Proteins, PLoS ONE 6(4): e18693, doi:10.1371/journal.pone.0018693 ArXiv:1104.3405

See Also

msr, AlexanderBriggs

Examples

## Not run: 
# link <- makeExampleKnot(k = FALSE)
# mVA(points3D = link$points3D, ends = link$ends)
# mVA(points3D = link$points3D, ends = link$ends, normalized = TRUE)
# mVA(points3D = link$points3D, ends = link$ends, normalized = TRUE, return.A = TRUE)
# ## End(Not run)