Bounding the Braid Index: Investigating the Morton Franks Williams Inequality in Knot Theory

In knot theory, a knot may have an invariant which is easily computed but difficult to understand geometrically and another invariant which is easily understood but quite difficult to compute. Knot theorists attempt to relate these two types of invariants. The Morton-Franks-Williams Inequality relates the easily understood braid index to the HOMFLY polynomial, which is straightforward to compute. We investigate the definition, theorems, and topics essential to understand the inequality, its implications, and the proof presented by J. Franks and R.F. Williams. Additionally, we prove that the Morton-Franks-Williams Inequality is sharp for all twist knots