Prove the Homfly polynomial of a link is determined by its Skein relations One of the questions from a past paper on my Knot Theory course is to prove that $P(L)$ (the Homfly polynomial of a link $L$) is determined by:
$$
(1) \ \ \ P(U_1)=1\\
(2) \ \ \ q^{-1}P(L_+)-qP(L_-)=zP(L_0)
$$
We may assume without proof that $P(U_n)$ is determined by $(1)$ and $(2)$ and that every link diagram may be changed into an unlink diagram by changing some crossings.
The result seems fairly obvious to me from assuming these things but the questions is worth a lot of marks, which makes me think I don't understand what the question requires. Does anyone have any insight?
 A: Usually the phrase "is determined by" means it is a statement about uniqueness, rather than one about existence.  However, in an examination I would ask for clarification.
Uniqueness is rather straightforward. You can argue by induction on a certain complexity measure for a link diagram $D$: $c(D)+u(D)$, where $c(D)$ is the number of crossings in the diagram, and $u(D)$ is the minimum number of crossings to change to get a diagram for an unlink.  They give you the base case for free.  The inductive case is that there is a crossing that can be changed, and (2) implies that the polynomial can be calculated by changing a crossing (decreasing $u(D)$) and smoothing a crossing (decreasing $c(D)$).  In particular, if $P_1$ and $P_2$ are two HOMFLY polynomials, then by induction $P_1$ and $P_2$ are the same for the diagrams with the changed and smoothed crossings, and so they are the same for the diagram under consideration.  Finally, every link has a diagram.
Just to give some flavor for the difficulty of existence, consider the following formulation.  Let $R=\mathbb{Z}[q^{\pm 1},z^{\pm 1}]$ be the ring of Laurent polynomials in the two variables $q$ and $z$, and let $M$ be the $R$-module generated by isotopy classes of links, modulo the skein relation (2).  You can extract from the argument of uniqueness that for every link $L$, there is some $P(q,z)\in R$ such that $[L]=P(q,z)[U_1]$, so not only is $M$ finitely generated, but $M=R[U_1]$, hence $M\cong R/\operatorname{Ann}_R([U_1])$.  A priori, $P(q,z)$ is well-defined up to an element of $\operatorname{Ann}_R([U_1])$. Existence is the claim that the ideal $\operatorname{Ann}_R([U_1])=\{p(q,z)\in R:p(q,z)[U_1]=0\}$ is trivial.
If I remember correctly, in Lickorish's An introduction to knot theory, he proves existence by showing by induction there is a well-defined HOMFLY polynomial defined for link diagrams with up to $n$ crossings, where the link diagrams are annotated with additional data that gives an unknotting order.  Then he shows that this choice of data does not matter.
