linkingNumber: Compute the linking number of a polygonal link
Description
Compute the linking number of a polygonal link
Usage
linkingNumber(points3D, ends, M = c())
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
M
the intersection matrix of the polygonal link. If no matrix is provided, the function will
compute it (default)
Value
lkthe linking number of the polygonal link
Details
The linking number is defined for a two-component oriented link as the sum of +1 crossings and -1
crossing over all crossings between the two links divided by 2. For components $\alpha$ and $\beta$,
$$lk(\alpha, \beta) = \frac{1}{2} \sum_{c \in \alpha \cap \beta} \epsilon(c)$$
where $\alpha \cap \beta$ is the set of crossings of $\alpha$ with $\beta$,
and $\epsilon(c)$ is the sign of the crossing.
References
Weisstein, Eric W. "Linking Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LinkingNumber.html
Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, p. 19, 1991.