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• For the algebra structure on H, we use the monoidal structure on the functor Φ: consider
which is an element in
m U,V : Φ(U) ⊗ Φ(V ) ∼ = Φ(U ⊗ V ) δ U⊗V
−→ Φ(U ⊗ V ) ⊗ H
∼ = Φ(U) ⊗ Φ(V ) ⊗ H
Nat (Φ 2 , Φ 2 ⊗ H) ∼ = Hom(H ⊗ H, H) .
This gives an associative product with unit element
η : K ∼ = Φ(I)
δ I
−→ Φ(I) ⊗ H ∼ = H .
• In a similar way, one uses the duality on C to obtain an antipode on H and shows that
H becomes a Hopf algebra.
• One finally shows that H−mod ∼ = C as monoidal categories.
✷
Remark 5.2.4.
Deligne has characterized [D, Theorem 7.1] those K-linear semisimple ribbon categories over
a field K of characteristic zero that admit a fibre functor: these are those categories for which
all objects have categorical dimensions that are non-negative integers.
5.3 Knots and links
Definition 5.3.1
1. A link (German: Verschlingung) in R 3 is a finite set of disjoint smoothly embedded circles
(without parametrization and orientation).
2. A link with a single component is called a knot.
3. An isotopy of a link is a smooth deformation of R 3 which does not induce intersections
and self intersections.
4. A framed link is a link with a non-zero normal vector field.
Links in the topological field theories of our interest are framed oriented links.
Examples 5.3.2.
1. A special example is the so-called unknot which is given by the unit circle in the x-y-plane
of R 3 .
2. Other important examples of well-known knots and links:
trefoil knot
Hopf link
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