# Trouble in understanding the proof of the existense of the HOMFLY polynomial

Following Page 252-254 of Christian Kassel's text "Quantum Groups" Chapter 10 very closely:

$$Definition$$ $$4.1$$

A triple $$(L^+,L_-,L_0)$$ of oriented links in $$\mathbb{R}^3$$ is a Conway triple if they can be represented by link Diagrams $$D_+,D_-,D_0$$ which coincide outside a disk in $$\mathbb{R}^2$$ and which are respectively isotopic to $$(X_+),(X_-),$$ and $$(||)$$, respectively, inside the disc:

(Enter here a picture of the standard diagram of skein relations for a polynomial knot invariant)

$$Theorem$$ $$4.2$$

$$\exists !$$ map $$L \rightarrow P_L$$ from the set of all oriented links in $$\mathbb{R}^3$$ to the ring $$\Delta=\mathbb{Z}[x,x^{-1},y,y^{-1}]$$ of two variable Laurent polynomials such that:

$$(i)$$: If $$L \cong L'$$, then $$P_L=P_{L'}$$

$$(ii)$$: The value of $$P$$ on the trivial knot is $$1$$

$$(iii)$$: Whenever $$(L_+,L_-,L_o)$$ is a conway triple we have:

$$xP_{L_x}-x^{-1}P_{L_-}=yP_{L_0} (*)$$

The invariant $$P_L$$ is called the Jones-Conway polynomial, or the HOMFLY polynomial.

Let $$K$$ be the set of isotopy equivalence classes of all oriented links in $$\mathbb{R}^3$$, and let $$\Delta[K]$$ be the free $$\Delta$$-module generated by $$K$$. Denote $$\Gamma$$ to be the quotient of $$\Delta[K]$$ by:

$$x[L_+]-x^{-1}[L_-]-y[L_0]$$

where $$(L_+,L_-,L_0)$$ runs over all Conway triples of $$K$$. The $$\Delta$$-module $$\Gamma$$ is called the skein module of $$\mathbb{R}^3$$.

$$Proposition$$ $$4.4$$: Let $$Q: \Delta \rightarrow \Gamma$$ be the $$\Delta$$-linear map sending $$1$$ to

the class $$[0]$$ to the trivial knot. Then Q is an isomorphism.

As a consequence of $$Proposition$$ $$4.4$$, the skein module $$\Gamma$$ is a free $$\Delta$$-module of rank one generated by $$[0]$$, the equivalence class of the trivial knot.

Proposition $$4.4$$ implies Theorem $$4.2$$. To see this, let $$[L] \in \Gamma$$ and set $$P_L=Q^{-1}([L])$$.

So we just need to prove Proposition $$4.4$$. We need to show that $$Q$$ is surjective and injective.

To show that $$Q$$ is surjective, it suffices to show that the $$\Delta$$-module $$\Gamma$$ is generated by the class $$[0]$$ of the trivial knot.

I can't understand the following proof of lemma $$4.5$$.

$$Lemma$$ $$4.5$$: The $$\Delta$$-module $$\Gamma$$ is generated by the family $$\{[O^{\otimes n}]\}_{n>0}$$ of isotopy classes of all trivial links.

$$proof$$: Let $$\Gamma_m$$ be the $$\Delta$$-submodule generated by the isotopy classes of links reprsentable by link diagrams with $$\leq m$$ crossing points. Clearly, $$\Gamma_{m}$$ maps to $$\Gamma_{m+1}...$$.

(What the heck is the author talking about here? What map is he talking about that maps $$\Gamma_{m}$$ maps into $$\Gamma_{m+1}$$? Surely not $$Q$$!!! $$Q$$ is not an endomorphism of $$\Gamma$$!!)

The proof then continues...

.. and $$\Gamma$$ is the union of $$\Gamma_m$$ and so it suffices to prove lemma 4.5 for each $$\Gamma_m$$. We proceed by induction, the base case is trivial. Suppose lemma 4.5 is true when there for all integers $$. Let $$[L] \in \Gamma_m$$. It may be represented by a link diagram wtih $$m$$ crossing points. Consider one of them. Then there exists a conway triple $$(L_+,L_-,L_0)$$ such that $$L=L_+$$ or $$L=L_-$$ and the diagram $$L_0$$ has less than $$m$$ crossing points. It follows from $$(*)$$ that $$[L_+]=x^{-2}[L_-]$$ mod $$\Gamma_{m-1}$$. In other words, a change of crossings changes the class of $$L$$ mod $$\Gamma_{m-1}$$ by multiplication by an invertible element of $$\Delta$$. Lemma 3.3 (stated below) implies that the class of $$L$$ belongs to the submodule generated by the trivial links and $$\Gamma_{m-1}$$. The latter is also generated by trivial links in view of the induction hypothesis.

(I also don't understand the bold part above, how does Lemma 3.3 imply this? Thanks!!)

$$Lemma$$ $$3.3$$: Any link diagram may be turned after appropriate changes of crossings into a link diagram representing a trivial link in $$\mathbb{R}^3$$

1. Any diagram of $$\leq m$$ crossings is also a diagram of $$\leq m+1$$ crossings, so $$\Gamma_m\hookrightarrow\Gamma_{m+1}$$ by sending generators over to generators and extend using $$\Delta$$-linearity.
2. Start with a diagram of $$m$$ crossings. We know there is a sequence of crossing changes that result in the trivial link (of $$\ell$$ circles). We use $$[L_+]=x^{-2}[L_-]\pmod{\Gamma_{m-1}}$$ to change one crossings, then another would be mod $$\Gamma_{m-2}$$ by multiplying by $$x^{\pm2}$$, etc. until you get to the trivial link. But $$\Gamma_i$$ are nested, so you can take everything mod $$\Gamma_{m-1}$$ and hence $$[L]\equiv x^{2k}[O^{\otimes \ell}]\pmod{\Gamma_{m-1}}$$, some integer $$k$$.