Description Usage Arguments Details Value References See Also Examples

View source: R/boundary_calculations.R

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

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