Skein based invariants and the Kauffman polynomial

This thesis uses Kauffman skein theory to give several new results. We show a correspondence between Kauffman and Homily satellite invariants with coefficients modulo 2, when we take certain patterns from the respective skeins of the annulus. Using stacked tangles we construct a polynomial time algorithm for calculating the Kauffman polynomial of links, and then extend the theory to give a new polynomial time algorithm for calculating the Homfly polynomial. We show that the Kauffman polynomials of genus 2 mutants can differ, and improve on existing examples showing the non-invariance of the Homfly polynomial under genus 2 mutation. By expressing twists as single crossings and smoothings in the Kauffman skein we develop an algorithm for calculating the Kauffman polynomial of pretzel links. Finally we consider the result of some calculations in the Kauffman skein of the annulus.