pdf file

link
= −(a 2 + a −2 )
link
= a
+ a −1
The vector space E(a) is called the skein module. (skein is in German “Gebinde”.) The class
of a link diagram D determines a vector 〈D〉(a) ∈ E(a).
Theorem 5.3.5.
1. The skein module is one-dimensional, dim C E(a) = 1 and canonically identified with C.
2. The skein class of a link is invariant under the Reidemester moves Ω ±1
0 , Ω ±1
2 , Ω ±1
3 and thus
an isotopy invariant of links.
Proof.
1. The Kauffman relations are sufficient to unknot any knot. The unknot is the identified
with the complex number −a 2 − a −2 .
2. To show invariance under Ω 0 , we compute:
= a + a −1 = (a(−a 2 − a −2 ) + a −1 ) = − a 3
In a similar way, we show for the opposite curl:
= − a −3
We conclude invariant under the Reidemeister move Ω ±1
0 .
3. Invariance under the Reidemeister move Ω ±1
2 is shown by a similar computation:
= a + a −1 =
= a 2 + + (−a 3 )a −1 =
128