boundary_matrix: This function calculates a boundary matrix.
In quhomology: Calculation of Homology of Quandles, Racks, Biquandles and Biracks

Description Usage Arguments Details Value References See Also Examples

This function calculates the boundary matrix for rack/birack for both the quandle and rack homology case. In particular, this is a representation of the boundary function in the simplicial complex of the rack/birack.

1	boundary_matrix(degree, k, degenerate = FALSE)

`degree`	This is the degree of the Homology group, that is, if one wants to calculate $H_3$, then degree=3. A positive integer.
`k`	This describes the order of the underlying rack or birack. A positive integer.
`degenerate`	If degenerate=TRUE, this calculates the boundary matrix for the quandle homology. If FALSE, the boundary matrix for the rack homology case is returned.

This functions takes all words (or just the non-degenerate ones) of length $degree$ in the rack/biquandle (which are represented by $Z_k$) and then calculates their boundary via the following equation. For this, let $x=(x_i)_0^degree-1$ be an element of the rack/birack and let $n:=degree-1$. $$partial(x) = Sum_i=0^n (-1)^i ( (x_0...(^x_i)...x_n)-(x_0^x_ix_1^x_i...x_i-1^x_ix_i+1_x_i...x_n_x_i) )$$, where ^x_i means except x_i. If this is a rack rather than a birack, remember that $f_a()=Id$.

A Matrix.

http://www.maths.sussex.ac.uk/Staff/RAF/Maths/homo.pdf

link{boundary_matrix_degenerate}

1	boundary_matrix(3,3,TRUE)

      [,1] [,2] [,3] [,4] [,5] [,6]
 [1,]   -1    1    0    1    0    0
 [2,]   -1    1    0    0   -1    0
 [3,]    1   -1    0   -1    0    0
 [4,]    1   -1    0    0    1    0
 [5,]    0    0   -1    1    0    1
 [6,]   -1    0   -1    1    0    0
 [7,]    0    0    1   -1    0   -1
 [8,]    1    0    1   -1    0    0
 [9,]    0    1    0    0   -1    1
[10,]    0    0   -1    0   -1    1
[11,]    0   -1    0    0    1   -1
[12,]    0    0    1    0    1   -1