What is a knot? Imagine a string, which we tie arbitrarily, and then fuse its two free ends together to form a closed loop. In technical language, a knot K is an embedding f:S^1\to\mathbb{R}^3. This paper investigates the first polynomial knot invariant, the Alexander polynomial, introduced by the American mathematician James Waddell Alexander II in 1923. We examine the Alexander polynomial of torus knots via two computing methods and the concrete form of this polynomial for torus knots. We then compare these polynomials and show the uniqueness of the Alexander polynomial for each torus knot, up to mirror images. Finally, we conclude with a Theorem from Stoimenow that the Alexander polynomial of a closed 3-braid is never 1, and prove it for four or less terms in the braid word.