Tangles and Categories of Tangles

Topological Definition of Tangles

Tangle

A tangle with bottom endpoints and top endpoints is an embedding such that it forms a bijection between the boundary of and the marked endpoints on bottom and top.

Stacking of Tangles

There are two ways to stack tangles. One is the horizontal (side by side) stacking, denoted by , and the other is the vertical stacking (composition), denoted by . The vertical stacking is he same associative stacking operation as for braids. Of course, two tangles may only be composed when the number of top endpoints of the first matches the number of bottom endpoints of the second. The horizontal stacking is then just put two tangles side by side. To simplify our terminology, we will refer to the vertical stacking as composition, and the horizontal stacking as stacking.

Why horizontal stacking?

Our goal is to write a finite presentation for tangles: importantly, a finite number of morphisms which generate all others. With only vertical stacking as the operation, this is not possible: essentially, morphisms between few endpoints don’t generate morphisms between many endpoints.

Tangle generators

Just follow from braids, there are infinitely many generators for tangles w.r.t composition: While, the number of generators for tangles w.r.t stacking is finite: We call them basic tangles, denoted as respectively.

e.g. We can generate the trefoil as a composition of basic tangles:

The Category of Tangles

Braided Monoidal Category

A braided monoidal category is a monoidal category with an additional structure, a braiding, which is a natural isomorphism that is compatible with the associativity and unit constraints.

Symmetric Monoidal Category

A symmetric monoidal category is a braided monoidal category with the braiding satisfying the symmetry axiom: .

Tangle Category

The category of tangles, denoted by , is a braided monoidal category consists of non-negative integers as objects, and tangles as morphisms, such that

• ,
• the composition of morphisms is tangle composition,
• the tensor product of morphisms is (horizontal) stacking,
• tensor product of objects is addition, is the monoidal unit.
• the braiding is the mirror of the tangle:

Remark

The braiding is not symmetric! Because the mirror of a tangle is not the same as the flipping of a tangle.

Proposition

is finitely presented.

Proof Objects generate all objects. And the morphisms (tangles) are generated by basic tangles under the following relations:

e.g. One of the first relation is of the form:

Oriented Tangles

Oriented Tangle

A oriented tangle is a tangle with a fixed orientation on each strand, denoted as .

Oriented Tangle Category

The category of oriented tangles, denoted by , is a braided monoidal category consists of

• words in as objects, (e.g. )
• oriented tangles as morphisms, such that the orientation of the tangle is compatible with the “orientation” of the objects.
• the composition of morphisms is tangle composition when the words match
• the tensor product of morphisms is (horizontal) stacking

Proposition

is finitely presented.

Proof The objects are generated by . The morphisms are like morphisms in but in all possible orientations.