checkRd: (-1) boundary_matrix.Rd:28: Lost braces; missing escapes or markup?
    28 | 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 $n:=degree-1$.
       |                                                                                                                                                                                                                                    ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                    ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                            ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                 ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                          ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                    ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                          ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                               ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                  ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                         ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                                 ^
checkRd: (-1) boundary_matrix.Rd:29: Lost braces; missing escapes or markup?
    29 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                                       ^
checkRd: (-1) boundary_matrix_degenerate.Rd:25: Lost braces; missing escapes or markup?
    25 | 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 $n:=degree-1$.
       |                                                                                                               ^
checkRd: (-1) boundary_matrix_degenerate.Rd:25: Lost braces; missing escapes or markup?
    25 | 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 $n:=degree-1$.
       |                                                                                                                                                                                                                ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                    ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                            ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                 ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                          ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                    ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                          ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                               ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                  ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                         ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                                 ^
checkRd: (-1) boundary_matrix_degenerate.Rd:26: Lost braces; missing escapes or markup?
    26 | $$partial(x) = Sum_{i=0}^n (-1)^i ( (x_0...{(^x_i)}...x_n)-(x_0^{x_i}x_1^{x_i}...x_{i-1}^{x_i}{x_{i+1}}_{x_i}...{x_n}_{x_i}) )$$, where ^x_i means except x_i.
       |                                                                                                                       ^
checkRd: (-1) boundary_names.Rd:26: Lost braces; missing escapes or markup?
    26 | This calculates all possible permutations of elements in $Z_k$ of length $degree$.  If degenerate is true, it loops through all of them, removing the degenerate ones (that is, those where $x_i=x_{i+1}$, for an element $x=(x_i)_0^{degree})$).
       |                                                                                                                                                                                                    ^
checkRd: (-1) boundary_names.Rd:26: Lost braces; missing escapes or markup?
    26 | This calculates all possible permutations of elements in $Z_k$ of length $degree$.  If degenerate is true, it loops through all of them, removing the degenerate ones (that is, those where $x_i=x_{i+1}$, for an element $x=(x_i)_0^{degree})$).
       |                                                                                                                                                                                                                                      ^
checkRd: (-1) boundary_names_degenerate.Rd:23: Lost braces; missing escapes or markup?
    23 | This calculates all possible permutations of elements in ${Z}_k$ of length $degree$.  If degenerate is true, it loops through all of them, removing the non-degenerate ones (that is, those where $x_i =/= x_{i+1}$ for all $i=0,...,degree-1$, for an element $x=(x_i)_0^{degree})$).
       |                                                           ^
checkRd: (-1) boundary_names_degenerate.Rd:23: Lost braces; missing escapes or markup?
    23 | This calculates all possible permutations of elements in ${Z}_k$ of length $degree$.  If degenerate is true, it loops through all of them, removing the non-degenerate ones (that is, those where $x_i =/= x_{i+1}$ for all $i=0,...,degree-1$, for an element $x=(x_i)_0^{degree})$).
       |                                                                                                                                                                                                              ^
checkRd: (-1) boundary_names_degenerate.Rd:23: Lost braces; missing escapes or markup?
    23 | This calculates all possible permutations of elements in ${Z}_k$ of length $degree$.  If degenerate is true, it loops through all of them, removing the non-degenerate ones (that is, those where $x_i =/= x_{i+1}$ for all $i=0,...,degree-1$, for an element $x=(x_i)_0^{degree})$).
       |                                                                                                                                                                                                                                                                           ^