Colorings beyond Fox: the other linear Alexander quandles
Louis H. Kauffman, Pedro Lopes

TL;DR
This paper explores advanced linear algebraic colorings in knot theory, extending beyond Fox colorings, and provides lower bounds for the minimal number of colors needed in these generalized colorings.
Contribution
It introduces new generalizations of linear Alexander quandle colorings and establishes lower bounds for the minimal number of colors required.
Findings
Lower bounds for the number of colors in generalized colorings
Extension of p-colorability to broader linear algebraic systems
Insights into the complexity of calculating minimal colors for larger primes
Abstract
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…
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