# nLab Shum's theorem

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Statement

###### Proposition

(Shum's theorem)
The category of framed oriented tangles $FrOrTang$ is equivalently the free ribbon category generated by a single object.

(Shum 1994, Yetter 2001 Thm. 9.1)

The following table indicates how the main axioms for ribbon categories translate to the corresponding moves on (framed, oriented) tangle/link diagrams (cf. e.g. Selinger 2011, Baez & Stay 2011), namely to isotopy moves and Reidemeister moves:

(NB: For the sliding moves, the corresponding mirrored moves are not shown, to save space.)

## Relation to knot invariants

This means that for $\mathcal{R}$ any ribbon category and $V \in \mathcal{R}$ any object, there is a unique ribbon functor

(1)$I^{\mathcal{R}}_V \;\colon\; FrTang \longrightarrow \mathcal{R}$

which sends the generating object $1 \in FrTan$ to $V \in \mathcal{R}$.

Since the endomorphisms of the object $0 \in FrOrTang$ are the oriented framed links this means that the functor $I^{\mathcal{R}}_V$ (1) restricts on endomorphisms to a collection of link invariants (with values in the endomorphism ring of $V$ whenever $\mathcal{R}$ is Ab-enriched).

The standard link invariants arise this way (e.g. the HOMFLY polynomial), with the ribbon category $\mathcal{R}$ being the category of representations (finite dimensional) of a quantum group (cf. Yetter 2001 p 118).

## References

Last revised on September 1, 2024 at 10:50:49. See the history of this page for a list of all contributions to it.