  This article is about applications of linear algebra to knot theory. For
example, for odd prime p, there is a rule (given in the article) for coloring
the arcs of a knot or link diagram from the residues mod p. This is a knot
invariant in the sense that if a diagram of the knot under study admits such a
coloring, then so does any other diagram of the same knot. This is called
p-colorability. It is also associated to systems of linear homogeneous
equations over the residues mod p, by regarding the arcs of the diagram as
variables and assigning the equation "twice the over-arc minus the sum of the
under-arcs equals zero" to each crossing. The knot invariant is here the
existence or non-existence of non-trivial solutions of these systems of
equations, when working over the integers modulo p (a non-trivial solution is
such that not all variables take up the same value). Another knot invariant is
the minimum number of distinct colors (values) these non-trivial solutions
require, should they exist. This corresponds to finding a basis, supported by a
diagram, in which these solutions take up the least number of distinct values.
The actual minimum is hard to calculate, in general. For the first few primes,
less than 17, it depends only on the prime, p, and not on the specific knots
that admit non-trivial solutions, modulo p. For primes larger than 13 this is
an open problem. In this article, we begin the exploration of other
generalizations of these colorings (which also involve systems of linear
homogeneous equations mod p) and we give lower bounds for the number of colors.
