Rknots (version 1.3.0)

intersectionMatrix: Compute the intersection matrix of a polygonal link

Description

Compute the intersection matrix of a polygonal link. See details.

Usage

intersectionMatrix(points3D, ends = c())

Arguments

points3D

an $N$ x 3 matrix of the $x$, $y$, $z$ coordinates of a three-dimensional structure

ends

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

Value

• An $N$-1 x $N$-1 matrix

Details

The entries of the intersection matrix are defined as follows. Given two sets of edges $A$ and $B$ we can compute the intersection matrix $I=I(A,B)$ by setting $$(I(A,B))_{i,j} = 0$$ if $A_i$ and $B_j$ do not intersect transversally.$$(I(A,B))_{i,j} = +1$$ if $A_i$ lays over $B_j$.$$(I(A,B))_{i,j} = -1$$ if $A_i$ lays under $B_j$. Finally, if $A=B$ we get the skew symmetric square matrix $I(A)$

References

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

Examples

##Compute the intersection matrix of a random structure of 20 points
points <- matrix(runif(60,-1,1), ncol = 3)

intersectionMatrix(points)

##Compute the intersection matrix of the trefoil knot
data(Rolfsen.table, package = "Rknots")
trefoil <- Rolfsen.table$"3.1"

intersectionMatrix(trefoil)