In this MSc thesis, which deals with certain topics from knot theory, we will engage with the problem of determining knot equivalences. More accurately, with polynomial invariants of knots, which map knots to certain polynomials.
With the aid of various skein relations and the writhe, we will define the following polynomial invariants, which will help us determine which knots are equivalent: Alexander polynomial, Alexander-Conway polynomial, Jones polynomial, Kauffman polynomial F, bracket polynomial, Kauffman polynomial X and HOMFLY-PT polynomial.
For every defined polynomial invariant, we will explicity compute its value for the Hopf link and the trefoil knot.
Some of these invariants dominate other invariants (HOMFLY-PT and Kauffman polynomial F), which means that they can distinguish between more knots than other invariants.
In this MSc thesis we will prove this preposition and find a complete set of relations between the defined polynomial invariants of knots.
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