boundary_matrix_degenerate: Calculation of boundary matrix for degenerate Homology.
In quhomology: Calculation of Homology of Quandles, Racks, Biquandles and Biracks

Description Usage Arguments Details Value References See Also Examples

This function returns the boundary matrix of a rack/birack necessary to calculate the degenerate Homology of the same. In particular, this is a representation of the boundary function in the simplicial complex of the rack/birack.

1	boundary_matrix_degenerate(degree, k)

`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.

This functions takes all degenerate words of length $degree$ in the rack/biquandle (which are represented by $Z_k$) and then calculates their boundary via the followi ng 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

boundary_matrix

1	boundary_matrix_degenerate(3,3)

      [,1] [,2] [,3]
 [1,]    0    0    0
 [2,]    0    0    0
 [3,]   -1    0    1
 [4,]   -1    1    0
 [5,]    0    0    0
 [6,]    0    0    0
 [7,]    0    0    0
 [8,]    1   -1    0
 [9,]    0   -1    1
[10,]    0    0    0
[11,]    0    0    0
[12,]    0    0    0
[13,]    0    1   -1
[14,]    1    0   -1
[15,]    0    0    0