# The Singular Temperley-Lieb Category

Authors: Carmen Caprau; Joel Smith
Journal: ISRN Geometry
 (2014)

---

## Abstract

We introduce and study the singular Temperley-Lieb category overℤ
[
q
,
q
-
1
], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for the s
l
(
2
) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.


---

## Body

## 1. Introduction

The Temperley-Lieb algebra has played a central role in the discovery of the Jones polynomial [1] and in the subsequent developments relating to knot theory, topological quantum field theory, and statistical mechanics [2]. Originally presented in terms of abstract generators and relations, it was combinatorially described by Kauffman as planar diagram algebra in terms of hisbracket polynomial. The Temperley-Lieb category provides a more structured perspective on the Temperley-Lieb algebra (for details we refer the reader to [3, 4]).Khovanov interpreted the Jones polynomial as the Euler characteristic of a cohomology theory of a link, now so-called the Khovanov homology [5]. In [6], Khovanov extended his homology theory to tangles, by associating a bimodule to morphism in the Temperley-Lieb category and a chain complex of bimodules to a plane diagram of a tangle. The chain homotopy equivalence class of the complex is an invariant of the tangle and specializes in the Khovanov homology when the tangle is a link. We remark that Bar-Natan [7] used another approach to Khovanov homology for tangles via cobordisms modulo local relations.The Khovanov homology satisfies the functoriality property under link cobordisms only up to a sign, but it was showed by Clark-Morrison-Walker [8], and independently by the first author [9] that there is a way to resolve the sign indeterminacy in the functoriality of the Khovanov homology. In [9], one uses webs and seamed cobordisms to construct a homology theory for tangles that is properly functorial under tangle cobordisms, and which recovers the Khovanov homology. In particular, one starts from a state model for the s
l
(
2
) link invariant, which is a state summation model for the unnormalized Jones polynomial (with t
1
/
2
=
-
q) via webs, which are oriented bivalent graphs whose vertices are either “sinks” or “sources.”This state model for thes
l
(
2
) polynomial for oriented links constitutes the motivation of this paper, and our main goal is to understand this model and the corresponding geometric objects from the category theory point of view. Regarding the vertices of the webs as singularities on diagrams, we call a state associated with a planar tangle diagram asingular flat tangle and we regard it as morphism in a monoidal category S
T
L, which we refer to as thesingular Temperley-Lieb category. It turns out that S
T
L is an autonomous category in which the objects are self-dual; that is, each object is isomorphic to its dual. The s
l
(
2
) link polynomial can be interpreted via representation theory of the quantum group U
q
(
s
l
(
2
)
). Specifically, an oriented edge in a singular flat tangle stands for the standard (2-dimensional) vector representation V
1 of quantum s
l
(
2
). Since V
1 is self-dual (i.e., isomorphic to its dual representation V
1
*), it is not surprising that every object in the singular Temperley-Lieb category S
T
L is self-dual.We hope that our paper will be a valuable resource for young researchers interested in knot invariants, in general, and categorical methods in representation theory and quantum invariants, in particular. Although we do not attempt it here, we remark that, as a byproduct of the construction of the singular Temperley-Lieb category, one can adapt Khovanov's work [6] to singular flat tangles and construct an s
l
(
2
) tangle (co)homology, which should be isomorphic to that developed in [9].The paper is organized as follows. We review thes
l
(
2
) polynomial in Section 2. Following Turaev [10], in Section 3 we briefly review the category of oriented tangles O
T
a and present it via generators and relations. We introduce and study the singular Temperley-Lieb category STL in Section 4. In Section 5 we construct a monoidal functor F
:
O
T
a
→
L
S
T
L, where L
S
T
L is an extension of S
T
L by allowing formal linear combinations of singular flat tangles with coefficients in ℤ
[
q
,
q
-
1
]. The functor F mimics the skein relations defining the s
l
(
2
) polynomial, and thus it can be regarded as a categorical approach to the link invariant. Finally, in Section 6 we construct a couple of polynomial invariants for oriented tangles, while in Section 7 we take another look at the category S
T
L by regarding a singular flat tangle as a directed ribbon graph.


## 2. A State Model for thes
l
(
2
) Polynomial

Thes
l
(
2
) polynomial can be defined as follows. Let D be a planar projection of an oriented link L and decompose each crossing of D as explained below
(1)We define〈
D
〉, the bracket of D, as the linear combination of the brackets of all resolutions Γ of D, where Γ is evaluated according the following rules:
(2)Specifically,〈
D
〉
=
∑
Γ
‍
(
±
q
α
(
Γ
)
〈
Γ
〉
), where the sum is over all resolutions (states) of D, and the signs ± and α
(
Γ
) are determined by the decomposition rules given in (1).It is easy to see thatq
2
〈
〉
-
q
-
2
〈
〉
=
(
q
-
q
-
1
)
〈
〉. It is known that 〈
D
1
〉
=
〈
D
2
〉 whenever D
1 and D
2 are ambient isotopic. This together with the above skein relations implies that 〈
D
〉 is the s
l
(
2
) link invariant or, equivalently, the unnormalized Jones polynomial of the oriented link L, with t
1
/
2
=
-
q.A resolutionΓ associated with the link diagram D is an s
l
(
2
) web, which is an oriented bivalent planar graph whose vertices are either “sources” or “sinks.” A resolution Γ might have a component with no vertices, that is, a closed oriented loop. For each vertex of an s
l
(
2
) web there is an ordering of the two adjacent edges meeting at that vertex, in the sense that one edge is the “preferred” edge of that vertex. Throughout the paper, we represent a bivalent vertex with a red triangle pointing toward the preferred edge of that vertex.


## 3. The Category of Oriented Tangles

An(
m
,
n
) tangle T is an embedding of a finite collection of arcs (homeomorphic to the interval [
0,1
]) and circles in ℝ
2
×
[
0,1
] such that the target n endpoints of T lie on ℝ
2
×
{
1
} and the source m endpoints of T lie on ℝ
2
×
{
0
}. Knots and links are (
0,0
) tangles and braids with n strands are the most well-known class of (
n
,
n
) tangles. The number of arcs in an (
m
,
n
) tangle is (
m
+
n
)
/
2; thus, m
+
n is required to be even.A diagram of an(
m
,
n
) tangle is a regular projection of the tangle into ℝ
×
[
0,1
]. Two tangles are e
q
u
i
v
a
l
e
n
t (or ambient isotopic) if one can be transformed into the other by an isotopy of ℝ
2
×
[
0,1
] which fixes the boundary.We will work with oriented tangles, that is, tangles whose components are oriented. IfT is an oriented (
m
,
n
) tangle, then the orientation of an arc in T induces orientations of its endpoints as follows: if an arc is oriented up (resp., down) near an endpoint of T, then the endpoint receives the positive orientation 1 (resp., the negative orientation -
1). Thus, to each oriented (
m
,
n
) tangle T we associate two sequences of numbers, (
a
1
,
a
2
,
…
,
a
m
) and (
b
1
,
b
2
,
…
,
b
n
), denoted source
T and target
T, respectively, where a
i
=
±
1 and b
i
=
±
1. If m
=
0 (resp., n
=
0), then source
T (resp., target
T) is the empty sequence ∅. In Figure 1 we provide an example of a (
4,4
) tangle diagram T with source
T
=
(
-
1,1
,
1
,
-
1
) and target
T
=
(
1
,
-
1
,
-
1,1
).Figure 1
An example of a(
4,4
) tangle.To avoid clutter, we will omit labeling the source and target of an oriented tangle, since these are clear from the orientations of tangle's components.There are two operations on tangles, namely, thetensor product  ⊗ and thecomposition  ∘, which are explained in Figure 2. The composition T
∘
T
′ is defined only when source
T
=
target
T
′.Figure 2
Tangle operations.Relations in the categoryO
T
a.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)We will denote the category of oriented tangles asO
T
a. The objects of this category are finite sequences of 1’s and −1’s, together with the empty sequence ∅. A m
o
r
p
h
i
s
m from A
=
(
a
1
,
a
2
,
…
,
a
m
) to B
=
(
b
1
,
b
2
,
…
,
b
n
) is an isotopy type of an oriented (
m
,
n
) tangle T with source
T
=
A and target
T
=
B. The category O
T
a is a monoidal (or tensor) category, where the tensor product of objects is the concatenation of sequences:
(3)
(
a
1
,
a
2
,
…
,
a
m
)
⊗
(
b
1
,
b
2
,
…
,
b
n
)
=
(
a
1
,
a
2
,
…
,
a
m
,
b
1
,
b
2
,
…
,
b
n
)
,
and the tensor product of morphisms is the tensor product of tangles. Observe that the empty sequence is theidentity object in O
T
a.Turaev [10] showed that the morphisms in O
T
a are generated under composition and tensor product by the following morphisms:
(4)
subject to the relations (
a
)–(
h
) listed below(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)where
(5)
These relations are illustrated in Figure 3.


## 4. The Singular Temperley-Lieb Category

Inspired by the state model for thes
l
(
2
) polynomial described in Section 2, we define thesingular Temperley-Lieb category, which we shall denote by S
T
L. The objects in S
T
L are the same as those in O
T
a, that is, finite sequences of 1’s and -
1’s, together with the empty sequence (which is the identity object in S
T
L). A morphism between objects A
=
(
a
1
,
a
2
,
…
,
a
m
) and B
=
(
b
1
,
b
2
,
…
,
b
n
) is an s
l
(
2
) web with boundary, considered up to planar isotopy, and can be regarded as a cobordism with source A and target B. We call such morphism asingular flat
(
m
,
n
) tangle, or asingular flat tangle. As before, m
+
n must be even, and we use the same convention as in the case of morphisms in O
T
a for labeling the boundary points of a singular flat tangle. An example of a (
6,4
)-singular flat tangle with source (
-
1,1
,
-
1
,
-
1,1
,
-
1
) and target (
1
,
-
1,1
,
1
) is depicted in Figure 4.Figure 4
A singular flat tangle.Observe that a singular flat(
m
,
n
) tangle is a resolution (state) corresponding to an oriented (
m
,
n
) tangle diagram T, obtained by applying the decomposition rules depicted in (1) to every crossing in T.


### 4.1. Generators and Relations

The composition and tensor product of singular flat tangles are defined in a similar manner as their analogues in the categoryO
T
a (see Figures 5 and 6).Figure 5
Composition of morphisms inS
T
L.Figure 6
Tensor product of morphisms inS
T
L.Figure 7
The generators forS
T
L.It is easy to see that the singular Temperley-Lieb categoryS
T
L is a monoidal category whose morphisms are generated under composition and tensor product by the morphisms depicted in Figure 7, subject to relations (6)–(15) given below(6)(7)(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)However, by the relations (6)–(11), we see that the only necessary generators are  and, say, A
+ and B
+.Relations (14) and (15) are depicted in Figure 8.Figure 8
The singularity canceling relations.We remark that the set of morphisms between objectsA and B in the category S
T
L is a ℤ
[
q
,
q
-
1
] module, where we mod out by the relations displayed in Figure 8, up to isotopy. We denote this space by Ho
m
S
T
L
(
A
,
B
).


### 4.2. More on the CategorySTL

In this section we study in more depth the singular Temperley-Lieb category and conclude that it is a pivotal category freely generated by two self-dual generators, namely, (1) and(
-
1
). For that, we need to recall some concepts from category theory.Let𝒞 be a monoidal category with identity object I. An object A in 𝒞 has adual, denoted by A
*, if there exists a pair of morphisms η
A
:
I
→
A
*
⊗
A and ϵ
A
:
A
⊗
A
*
→
I, such that
(16)
(
ϵ
A
⊗
id
A
)
∘
(
id
A
⊗
η
A
)
=
id
A
,
(
id
A
*
⊗
ϵ
A
)
∘
(
η
A
⊗
id
A
*
)
=
id
A
*
.If every object in𝒞 has a dual, then 𝒞 is called anautonomous (orcompact) category. The following holds in an autonomous category, for any two objects A and B:
(17)
(
A
⊗
B
)
*
=
B
*
⊗
A
*
.Thedual of morphism
α
:
A
→
B is a morphism α
*
:
B
*
→
A
* that satisfies the following two equations:
(18)
α
*
=
(
id
A
*
⊗
ϵ
B
)
∘
(
id
A
*
⊗
α
⊗
id
B
*
)
∘
(
η
A
⊗
id
B
*
)
,
α
=
(
ϵ
B
*
⊗
id
A
*
)
∘
(
id
B
*
⊗
α
⊗
id
A
*
)
∘
(
id
B
*
⊗
η
A
*
)
.It is well known that the categoryO
T
a of oriented tangles is an autonomous category, where (
1
)
*
=
(
-
1
) and (
-
1
)
*
=
(
1
). To see this, let η
1
=
and ϵ
1
=
, and observe that
(19)
are equivalent to (16) for the object A
=
(
1
) in O
T
a. Similarly, let η
-
1
=
and ϵ
-
1
=
, and recall that the following holds in O
T
a
(20)
which are pictorial representations for (16) corresponding to the object A
=
(
-
1
).Since the above relations hold in the categoryS
T
L as well, the following statement holds.Proposition 1.
The singular Temperley-Lieb categoryS
T
L is an autonomous category.Relations(
a
)-(
b
) in O
T
a and equivalently relations (12)-(13) in S
T
L imply that  and  are dual morphisms (of each other) in both categories, O
T
a and S
T
L. Moreover, in S
T
L the dual of  is , and the dual of  is . Therefore, the dual operation for morphisms inSTL switches the preferred side of a vertex.A morphismα
:
A
→
B in a category 𝒞 is said to be i
n
v
e
r
t
i
b
l
e if there exists a morphism β
:
B
→
A in 𝒞 such that α
∘
β
=
i
d
B and β
∘
α
=
i
d
A. In this case, β is said to be theinverse of α and is denoted by α
-
1. If a morphism α
:
A
→
B is invertible, then A is said to be i
s
o
m
o
r
p
h
i
c to B and denoted by A
≅
B. An invertible morphism is also called anisomorphism.Observe that relations (14)-(15) in S
T
L imply that the morphisms , , , and  are invertible, where
(21)
Therefore, (
1
)
≅
(
-
1
) in the category S
T
L, which implies that the objects in S
T
L areself-dual; that is, every object is isomorphic to its dual.Apivotal category is an autonomous category 𝒞 where (
A
*
)
*
≅
A, for every object A in 𝒞. Therefore, categories O
T
a and S
T
L are examples of pivotal categories. In fact, (
A
*
)
*
=
A, for every object A in O
T
a.


### 4.3.  The Linear Singular Temperley-Lieb Category

We define thelinear singular Temperley-Lieb category and denote it by L
S
T
L, as the free ℤ
[
q
,
q
-
1
]-linear category generated by S
T
L. The objects of L
S
T
L are the same as those of S
T
L, and the morphisms are ℤ
[
q
,
q
-
1
]-linear combinations of morphisms in S
T
L. The composition of morphisms in L
S
T
L is defined by bilinear extension of the composition of morphisms in S
T
L.This new category is subject to the same relations as inS
T
L, together with relations (22) depicted below
(22)
Relations (22) “remove” all circles and piecewise oriented circles with two bivalent vertices (whose red triangles point to each other), and replace them with q
+
q
-
1.Observe that the set of morphisms between objectsA and B in L
S
T
L is a ℤ
[
q
,
q
-
1
] module, which we denote by Ho
m
L
S
T
L
(
A
,
B
), where we mod out by the relations (22) and those displayed in Figure 8, up to isotopy.


## 4.1. Generators and Relations

The composition and tensor product of singular flat tangles are defined in a similar manner as their analogues in the categoryO
T
a (see Figures 5 and 6).Figure 5
Composition of morphisms inS
T
L.Figure 6
Tensor product of morphisms inS
T
L.Figure 7
The generators forS
T
L.It is easy to see that the singular Temperley-Lieb categoryS
T
L is a monoidal category whose morphisms are generated under composition and tensor product by the morphisms depicted in Figure 7, subject to relations (6)–(15) given below(6)(7)(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)However, by the relations (6)–(11), we see that the only necessary generators are  and, say, A
+ and B
+.Relations (14) and (15) are depicted in Figure 8.Figure 8
The singularity canceling relations.We remark that the set of morphisms between objectsA and B in the category S
T
L is a ℤ
[
q
,
q
-
1
] module, where we mod out by the relations displayed in Figure 8, up to isotopy. We denote this space by Ho
m
S
T
L
(
A
,
B
).


## 4.2. More on the CategorySTL

In this section we study in more depth the singular Temperley-Lieb category and conclude that it is a pivotal category freely generated by two self-dual generators, namely, (1) and(
-
1
). For that, we need to recall some concepts from category theory.Let𝒞 be a monoidal category with identity object I. An object A in 𝒞 has adual, denoted by A
*, if there exists a pair of morphisms η
A
:
I
→
A
*
⊗
A and ϵ
A
:
A
⊗
A
*
→
I, such that
(16)
(
ϵ
A
⊗
id
A
)
∘
(
id
A
⊗
η
A
)
=
id
A
,
(
id
A
*
⊗
ϵ
A
)
∘
(
η
A
⊗
id
A
*
)
=
id
A
*
.If every object in𝒞 has a dual, then 𝒞 is called anautonomous (orcompact) category. The following holds in an autonomous category, for any two objects A and B:
(17)
(
A
⊗
B
)
*
=
B
*
⊗
A
*
.Thedual of morphism
α
:
A
→
B is a morphism α
*
:
B
*
→
A
* that satisfies the following two equations:
(18)
α
*
=
(
id
A
*
⊗
ϵ
B
)
∘
(
id
A
*
⊗
α
⊗
id
B
*
)
∘
(
η
A
⊗
id
B
*
)
,
α
=
(
ϵ
B
*
⊗
id
A
*
)
∘
(
id
B
*
⊗
α
⊗
id
A
*
)
∘
(
id
B
*
⊗
η
A
*
)
.It is well known that the categoryO
T
a of oriented tangles is an autonomous category, where (
1
)
*
=
(
-
1
) and (
-
1
)
*
=
(
1
). To see this, let η
1
=
and ϵ
1
=
, and observe that
(19)
are equivalent to (16) for the object A
=
(
1
) in O
T
a. Similarly, let η
-
1
=
and ϵ
-
1
=
, and recall that the following holds in O
T
a
(20)
which are pictorial representations for (16) corresponding to the object A
=
(
-
1
).Since the above relations hold in the categoryS
T
L as well, the following statement holds.Proposition 1.
The singular Temperley-Lieb categoryS
T
L is an autonomous category.Relations(
a
)-(
b
) in O
T
a and equivalently relations (12)-(13) in S
T
L imply that  and  are dual morphisms (of each other) in both categories, O
T
a and S
T
L. Moreover, in S
T
L the dual of  is , and the dual of  is . Therefore, the dual operation for morphisms inSTL switches the preferred side of a vertex.A morphismα
:
A
→
B in a category 𝒞 is said to be i
n
v
e
r
t
i
b
l
e if there exists a morphism β
:
B
→
A in 𝒞 such that α
∘
β
=
i
d
B and β
∘
α
=
i
d
A. In this case, β is said to be theinverse of α and is denoted by α
-
1. If a morphism α
:
A
→
B is invertible, then A is said to be i
s
o
m
o
r
p
h
i
c to B and denoted by A
≅
B. An invertible morphism is also called anisomorphism.Observe that relations (14)-(15) in S
T
L imply that the morphisms , , , and  are invertible, where
(21)
Therefore, (
1
)
≅
(
-
1
) in the category S
T
L, which implies that the objects in S
T
L areself-dual; that is, every object is isomorphic to its dual.Apivotal category is an autonomous category 𝒞 where (
A
*
)
*
≅
A, for every object A in 𝒞. Therefore, categories O
T
a and S
T
L are examples of pivotal categories. In fact, (
A
*
)
*
=
A, for every object A in O
T
a.


## 4.3.  The Linear Singular Temperley-Lieb Category

We define thelinear singular Temperley-Lieb category and denote it by L
S
T
L, as the free ℤ
[
q
,
q
-
1
]-linear category generated by S
T
L. The objects of L
S
T
L are the same as those of S
T
L, and the morphisms are ℤ
[
q
,
q
-
1
]-linear combinations of morphisms in S
T
L. The composition of morphisms in L
S
T
L is defined by bilinear extension of the composition of morphisms in S
T
L.This new category is subject to the same relations as inS
T
L, together with relations (22) depicted below
(22)
Relations (22) “remove” all circles and piecewise oriented circles with two bivalent vertices (whose red triangles point to each other), and replace them with q
+
q
-
1.Observe that the set of morphisms between objectsA and B in L
S
T
L is a ℤ
[
q
,
q
-
1
] module, which we denote by Ho
m
L
S
T
L
(
A
,
B
), where we mod out by the relations (22) and those displayed in Figure 8, up to isotopy.


## 5. Thes
l
(
2
) Polynomial Revisited

We define a monoidal functorF
:
O
T
a
→
L
S
T
L from the category of oriented tangles to the linear singular Temperley-Lieb category. The functor F is the identity on objects. Moreover, the functor F is the identity on all of the generating morphisms for O
T
a, except for  and , for which we have
(23)The functorF is well defined, in the sense that all of the defining relations in O
T
a hold for their images under T in L
S
T
L. The planar relations (
a
) and (
b
) follow since these relations hold in both categories, and since F is the identity on the morphisms that are used to form these relations. The image under F of the first equality (
c
) equals
(24)
The second equality (
c
) is verified similarly. For relation (
d
), observe the following:
(25)Therefore, we have(26)
which verifies that the first part of relation (
d
) in O
T
a holds for its image in L
S
T
L. The second relation (
d
) is verified in a similar manner.Now we take a look at relation(
e
). The image of the left hand side of the equality equals(27)Applying similar computations to the right hand side of relation(
e
), we obtain
(28)
And, therefore, F
= F
.For relation(
f
), we have
(29)
as desired. Finally, we verify relation (
g
):
(30)
Relation (
h
) is verified similarly, and we leave it for the reader. Therefore, the functor F is well defined.Our construction implies that ifT is an oriented (
m
,
n
) tangle, then F
(
T
) is an ambient isotopy invariant for T. Moreover, an oriented link L is a (
0,0
) tangle, and F
(
L
) is an ambient isotopy invariant of L which equals the s
l
(
2
) polynomial of L. Thus, our functor F provides another way of defining the s
l
(
2
) link invariant.


## 6. Polynomial Invariants for Tangles

The scope of this section is to show how to use the geometric categoriesS
T
L and L
S
T
L to construct invariants for tangles.


### 6.1. The Case of(
n
,
n
) Tangle

In this section we restrict our attention to endomorphisms inO
T
a
,
S
T
L
, and L
S
T
L. Recall that anendomorphism in a category 𝒞 is morphism α
:
A
→
A whose source and target are the same object in 𝒞.Thetrace of an endomorphism α
:
A
→
A in a pivotal category with identity object I is the endomorphism of I given by the formula
(31)
tr
⁡
(
α
)
=
ϵ
A
∘
(
α
⊗
i
d
A
*
)
∘
η
A
*
.For any endomorphismα in S
T
L obtained by applying the functor F to a tangle in O
T
a, its trace, tr
⁡
(
α
), is a polynomial in ℤ
[
q
,
q
-
1
]. We regard tr
⁡ as a map on the space of endomorphisms En
d
S
T
L
(
A
), and extend it by bilinearity to En
d
L
S
T
L
(
A
).Example 2.
Consider the following:(32)Definition 3.
Given an(
n
,
n
) tangle T, define a polynomial P
(
T
)
∈
ℤ
[
q
,
q
-
1
] by
(33)
P
(
T
)
=
tr
⁡
(
F
(
T
)
)
.Proposition 4.
IfD is a diagram of an oriented (
n
,
n
) tangle T, then P
(
D
) is an ambient isotopy invariant for T.Proof.
Since the functorF
:
O
T
a
→
L
S
T
L is well defined, F
(
D
)
=
F
(
D
′
) whenever D and D
′ differ by a Reidemeister move. In this case, tr
⁡
(
F
(
D
)
)
=
tr
⁡
(
F
(
D
′
)
), and therefore, F
(
T
) is independent of the diagram D.Example 5.
Consider the oriented tangle diagramT
=
. Applying the functor F, we have the following linear combination of singular flat tangles:
(34)
Applying the trace function, we obtain
(35)
We have obtained that P() = 2
q
6
+
4
q
4
+
3
q
2
+
1.


### 6.2.  The Case of Arbitrary Tangles

We consider now arbitrary(
m
,
n
) tangles and construct a polynomial invariant for them.Given a singular flat tangleα, where α
:
A
→
B, we define the i
n
v
o
l
u
t
i
o
n of α as the singular flat tangle W
(
α
)
:
B
→
A obtained by reflecting α about the line ℝ
×
{
1
/
2
}, followed by reversing the orientations of the strands. The preferred side of a vertex is carried forward by the reflection operation. An example is depicted in Figure 9.Figure 9
InvolutionW.We regardW as a map on the space Ho
m
S
T
L
(
A
,
B
)
(36)
W
:
Ho
m
S
T
L
(
A
,
B
)
⟶
Ho
m
S
T
L
(
B
,
A
)
,
and extend it by linearity to Ho
m
L
S
T
L
(
A
,
B
).Definition 6.
Given a singular flat tangleα, define K
(
α
)
=
W
(
α
)
∘
α.Observe thatK
:
Ho
m
S
T
L
(
A
,
B
)
→
En
d
S
T
L
(
A
) and that it can be extended to Ho
m
L
S
T
L
(
A
,
B
) by bilinearity. Moreover, K
(
α
) is an endomorphism of A in S
T
L, and thus we can take the trace of it, tr
⁡
(
K
(
α
)
).Definition 7.
Given an oriented(
m
,
n
) tangle T, define the Laurent polynomial I
(
T
)
∈
ℤ
[
q
,
q
-
1
] by
(37)
I
(
T
)
=
tr
⁡
(
K
(
F
(
T
)
)
)
.The following statement follows at once from our construction.Proposition 8.
I
(
T
) is an ambient isotopy invariant for oriented tangles.Example 9.
Find the polynomialI
(
T
) for
(38)
Applying the functor F to T, we have
(39)
The involution tangles W
(
α
) of the singular flat tangles α found in the previous step are
(40)
Composing all singular flat tangles α obtained in the first step with their corresponding involution tangle W
(
α
), we obtain
(41)
Finally, applying the trace function and the relations (22), we have
(42)
Therefore, I
(
T
)
=
(
q
+
q
-
1
)
3
(
q
-
1
-
q
-
2
).


## 6.1. The Case of(
n
,
n
) Tangle

In this section we restrict our attention to endomorphisms inO
T
a
,
S
T
L
, and L
S
T
L. Recall that anendomorphism in a category 𝒞 is morphism α
:
A
→
A whose source and target are the same object in 𝒞.Thetrace of an endomorphism α
:
A
→
A in a pivotal category with identity object I is the endomorphism of I given by the formula
(31)
tr
⁡
(
α
)
=
ϵ
A
∘
(
α
⊗
i
d
A
*
)
∘
η
A
*
.For any endomorphismα in S
T
L obtained by applying the functor F to a tangle in O
T
a, its trace, tr
⁡
(
α
), is a polynomial in ℤ
[
q
,
q
-
1
]. We regard tr
⁡ as a map on the space of endomorphisms En
d
S
T
L
(
A
), and extend it by bilinearity to En
d
L
S
T
L
(
A
).Example 2.
Consider the following:(32)Definition 3.
Given an(
n
,
n
) tangle T, define a polynomial P
(
T
)
∈
ℤ
[
q
,
q
-
1
] by
(33)
P
(
T
)
=
tr
⁡
(
F
(
T
)
)
.Proposition 4.
IfD is a diagram of an oriented (
n
,
n
) tangle T, then P
(
D
) is an ambient isotopy invariant for T.Proof.
Since the functorF
:
O
T
a
→
L
S
T
L is well defined, F
(
D
)
=
F
(
D
′
) whenever D and D
′ differ by a Reidemeister move. In this case, tr
⁡
(
F
(
D
)
)
=
tr
⁡
(
F
(
D
′
)
), and therefore, F
(
T
) is independent of the diagram D.Example 5.
Consider the oriented tangle diagramT
=
. Applying the functor F, we have the following linear combination of singular flat tangles:
(34)
Applying the trace function, we obtain
(35)
We have obtained that P() = 2
q
6
+
4
q
4
+
3
q
2
+
1.


## 6.2.  The Case of Arbitrary Tangles

We consider now arbitrary(
m
,
n
) tangles and construct a polynomial invariant for them.Given a singular flat tangleα, where α
:
A
→
B, we define the i
n
v
o
l
u
t
i
o
n of α as the singular flat tangle W
(
α
)
:
B
→
A obtained by reflecting α about the line ℝ
×
{
1
/
2
}, followed by reversing the orientations of the strands. The preferred side of a vertex is carried forward by the reflection operation. An example is depicted in Figure 9.Figure 9
InvolutionW.We regardW as a map on the space Ho
m
S
T
L
(
A
,
B
)
(36)
W
:
Ho
m
S
T
L
(
A
,
B
)
⟶
Ho
m
S
T
L
(
B
,
A
)
,
and extend it by linearity to Ho
m
L
S
T
L
(
A
,
B
).Definition 6.
Given a singular flat tangleα, define K
(
α
)
=
W
(
α
)
∘
α.Observe thatK
:
Ho
m
S
T
L
(
A
,
B
)
→
En
d
S
T
L
(
A
) and that it can be extended to Ho
m
L
S
T
L
(
A
,
B
) by bilinearity. Moreover, K
(
α
) is an endomorphism of A in S
T
L, and thus we can take the trace of it, tr
⁡
(
K
(
α
)
).Definition 7.
Given an oriented(
m
,
n
) tangle T, define the Laurent polynomial I
(
T
)
∈
ℤ
[
q
,
q
-
1
] by
(37)
I
(
T
)
=
tr
⁡
(
K
(
F
(
T
)
)
)
.The following statement follows at once from our construction.Proposition 8.
I
(
T
) is an ambient isotopy invariant for oriented tangles.Example 9.
Find the polynomialI
(
T
) for
(38)
Applying the functor F to T, we have
(39)
The involution tangles W
(
α
) of the singular flat tangles α found in the previous step are
(40)
Composing all singular flat tangles α obtained in the first step with their corresponding involution tangle W
(
α
), we obtain
(41)
Finally, applying the trace function and the relations (22), we have
(42)
Therefore, I
(
T
)
=
(
q
+
q
-
1
)
3
(
q
-
1
-
q
-
2
).


## 7. Ribbon Graphs and Singular Flat Tangles

Aband is the image of the square [
0,1
]
×
[
0,1
] under its embedding in ℝ
3, and anannulus is the image of the cylinder S
1
×
[
0,1
] under its embedding in ℝ
3. The core of the band (equivalently the core of the annulus) is the image of the segment {
1
/
2
}
×
[
0,1
] (equivalently S
1
×
{
1
/
2
}) under this embedding.Letm and n be nonnegative integers with m
+
n even. Aribbon
(
m
,
n
) graph is an oriented surface S embedded in ℝ
2
×
[
0,1
] and decomposed into (
m
+
n
)
/
2 bands and a finite collection of annuli. The choice of orientation for S is equivalent to a choice of one side of S. Rotating a band (or an annulus) by 18
0
∘ around its core results in the same band (or annulus) with the opposite orientation.A ribbon graph is calleddirected if the cores of its bands and annuli are equipped with directions. Note that the cores of the bands and annuli of the surface S are provided with opposite directions on the two sides of S.With each directed ribbon(
m
,
n
) graph we associate two sequences (
a
1
,
a
2
,
…
,
a
m
) and (
b
1
,
b
2
,
…
,
b
n
) of ±
1 in a similar fashion as we did for oriented tangles and singular flat tangles.We can regard a singular flat(
m
,
n
) tangle α as a directed ribbon (
m
,
n
) graph S. Since a bivalent vertex in α is a singularity where orientations disagree, it corresponds to a half-twist in a band or an annulus of the ribbon graph S. If the red triangle corresponding to a bivalent vertex in α is oriented up or to the right, then the half-twist of the corresponding band/annulus is a right twist in the neighborhood of the vertex. Otherwise, it is a left twist:
(43)As explained in Section4.2, the above generators forSTL are isomorphisms, and thus (
1
)
≅
(
-
1
). It implies that the category S
T
L is self-dual, where there exists an isomorphism h
A
:
A
→
A
* for every object A (represented as a half-twist).ForA
=
(
1
) let h
(
1
)
=
, and for A
=
(
-
1
) let h
(
-
1
) = . Observe that if h
(
-
1
) =
⇝
, then its inverse is h
(
-
1
)
-
1 =
⇝
, and their composition is
(44)We remark that the self-duality imposes abraiding onSTL via the isomorphisms c
A
,
B
:
A
⊗
B
→
B
⊗
A given by
(45)
c
A
,
B
:
=
A
⊗
B
→
h
A
⊗
B
(
A
⊗
B
)
*
→
=
B
*
⊗
A
*
→
h
B
-
1
⊗
h
A
-
1
B
⊗
A
,
for any pair of objects A and B.Moreover, for every objectA in S
T
L, there is an isomorphism θ
A
*
:
A
*
→
A
* given by θ
A
*
:
=
h
A
*
∘
h
A
* and represented as a full twist on A
*. For example, if A
=
(
1
), then A
*
=
(
-
1
) and θ
A
* is represented as
(46)
Note that θ
A
*
=
(
θ
A
)
* and c
B
,
A
∘
(
θ
B
⊗
θ
A
)
∘
c
A
,
B
=
θ
A
⊗
B for any objects A
,
B in S
T
L, which implies that the category S
T
L istortile (see [11]).



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*Source: 321509-2014-04-24.xml*