A Connection Between Knot Theory and Linear Algebra

In knot theory literature, a lot of references treated the topic of computing the Alexander polynomial and proved that the Alexander polynomial is a knot invariant. In this paper, we concentrate on an unconventional connection between knot theory and linear algebra in computing the Alexander polynomial and introduce an approach depending linear algebra tools to show that the Alexander polynomial is a knot invariant. Also, we connect this approach to the way of Fox coloring to suggest another way to prove that the 3-coloring is a knot invariant and confirm what is already known about the 3-coloring of the trefoil and the figure eight knots.