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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 $<m$. 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 $\mathbb{R}^3$

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1 Answer 1

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  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 $k$.

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