The additivity of crossing number with respect to the composition of knots

Abstract

This paper will investigate the additivity of the crossing number with respect to the composition of knots. The additivity of the crossing number is a long standing conjecture. The paper presents proofs of this conjecture for alternating knots and torus knots. For alternating knots, the paper uses the Jones Polynomial to show the alternating diagram has minimal degree, and proves the composition of two alternating knots is another alternating knot. For torus knots, the paper’s main ingredient is a closed form equality for the crossing number involving the braid index and genus of the knot. We then show the additivity under composition of these components of the formula to prove the additivity of the crossing number.