The Singular Temperley-Lieb Category

We introduce and study the singular Temperley-Lieb category over Z[q, q], 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 sl(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.


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 his bracket 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 (2) link invariant, which is a state summation model for the unnormalized Jones polynomial (with  1/2 = −) via webs, which are oriented bivalent graphs whose vertices are either "sinks" or "sources." This state model for the (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 a singular flat tangle and we regard it as morphism in a monoidal category STL, which we refer to as the singular Temperley-Lieb category.It turns out that STL is an autonomous category in which the objects are self-dual; that is, each object is isomorphic to its dual.The (2) link polynomial can be interpreted via representation theory of the quantum group   ((2)).Specifically, an oriented edge in a singular flat tangle stands for the standard (2-dimensional) vector representation  1 of quantum  (2).Since  1 is selfdual (i.e., isomorphic to its dual representation  * 1 ), it is not surprising that every object in the singular Temperley-Lieb category STL 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 (2) tangle (co)homology, which should be isomorphic to that developed in [9].
The paper is organized as follows.We review the (2) polynomial in Section 2. Following Turaev [10], in Section 3 we briefly review the category of oriented tangles OTa 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  : OTa → LSTL, where LSTL is an extension of STL by allowing formal linear combinations of singular flat tangles with coefficients in Z[,  −1 ].The functor  mimics the skein relations defining the (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 STL by regarding a singular flat tangle as a directed ribbon graph.

A State Model for the 𝑠𝑙(2) Polynomial
The (2) polynomial can be defined as follows.Let  be a planar projection of an oriented link  and decompose each crossing of  as explained below We define ⟨⟩, the bracket of , as the linear combination of the brackets of all resolutions Γ of , where Γ is evaluated according the following rules: (2) Specifically, ⟨⟩ = ∑ Γ (± (Γ) ⟨Γ⟩), where the sum is over all resolutions (states) of , and the signs ± and (Γ) are determined by the decomposition rules given in (1).
A resolution Γ associated with the link diagram  is an (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 (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.

The Category of Oriented Tangles
An (, ) tangle  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  endpoints of  lie on R 2 × {1} and the source  endpoints of  lie on R 2 × {0}.Knots and links are (0, 0) tangles and braids with  strands are the most wellknown class of (, ) tangles.The number of arcs in an (, ) tangle is ( + )/2; thus,  +  is required to be even.A diagram of an (, ) tangle is a regular projection of the tangle into R × [0, 1].Two tangles are V (or ambient isotopic) if one can be transformed into the other by an isotopy of R 2 × [0, 1] which fixes the boundary.
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, the tensor product ⊗ and the composition ∘, which are explained in Figure 2. The composition  ∘   is defined only when source  = target   .
We will denote the category of oriented tangles as OTa.The objects of this category are finite sequences of 1's and −1's, together with the empty sequence 0. A ℎ from  = ( 1 ,  2 , . . .,   ) to  = ( 1 ,  2 , . . .,   ) is an isotopy type of an oriented (, ) tangle  with source  =  and target  = .The category OTa is a monoidal (or tensor) category, where the tensor product of objects is the concatenation of sequences: and the tensor product of morphisms is the tensor product of tangles.Observe that the empty sequence is the identity object in OTa.
Turaev [10] showed that the morphisms in OTa are generated under composition and tensor product by the following morphisms: , , , subject to the relations (a)-(h) listed below (a) These relations are illustrated in Figure 3.

The Singular Temperley-Lieb Category
Inspired by the state model for the (2) polynomial described in Section 2, we define the singular Temperley-Lieb category, which we shall denote by STL.The objects in STL are the same as those in OTa, that is, finite sequences of 1's and −1's, together with the empty sequence (which is the identity object in STL).A morphism between objects  = ( 1 ,  2 , . . .,   ) and  = ( 1 ,  2 , . . .,   ) is an (2) web with boundary, considered up to planar isotopy, and can be regarded as a cobordism with source  and target .We call such morphism a singular flat (, ) tangle, or a singular flat tangle.As before,  +  must be even, and we use the same convention as in the case of morphisms in OTa 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.
Observe that a singular flat (, ) tangle is a resolution (state) corresponding to an oriented (, ) tangle diagram , obtained by applying the decomposition rules depicted in (1) to every crossing in .

Generators and Relations.
The composition and tensor product of singular flat tangles are defined in a similar manner as their analogues in the category OTa (see Figures 5  and 6).
It is easy to see that the singular Temperley-Lieb category STL 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

B +
), However, by the relations ( 6)-( 11), we see that the only necessary generators are , , , , , , and, say,  + and  + .Relations ( 14) and (15) are depicted in Figure 8.We remark that the set of morphisms between objects  and  in the category STL is a Z[,  −1 ] module, where we mod out by the relations displayed in Figure 8, up to isotopy.We denote this space by Hom STL (, ).

More on the Category STL.
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.
Let C be a monoidal category with identity object .An object  in C has a dual, denoted by  * , if there exists a pair of morphisms   :  →  * ⊗  and   : ⊗ * → , such that If every object in C has a dual, then C is called an autonomous (or compact) category.The following holds in an autonomous category, for any two objects  and : The dual of morphism  :  →  is a morphism  * :  * →  * that satisfies the following two equations: It is well known that the category OTa of oriented tangles is an autonomous category, where (1) * = (−1) and (−1) * = (1).To see this, let  1 = and  1 = , and observe that are equivalent to (16) for the object  = (1) in OTa.Similarly, let  −1 = and  −1 = , and recall that the following holds in OTa which are pictorial representations for (16) corresponding to the object  = (−1).
Since the above relations hold in the category STL as well, the following statement holds.

Proposition 1. The singular Temperley-Lieb category STL is an autonomous category.
Relations (a)-(b) in OTa and equivalently relations (12)-(13) in STL imply that and are dual morphisms (of each other) in both categories, OTa and STL.Moreover, in STL the dual of is , and the dual of is .Therefore, the dual operation for morphisms in STL switches the preferred side of a vertex.
A morphism  :  →  in a category C is said to be V if there exists a morphism  :  →  in C such that  ∘  = id  and  ∘  = id  .In this case,  is said to be the inverse of  and is denoted by  −1 .If a morphism  : →  is invertible, then  is said to be ℎ to  and denoted by  ≅ .An invertible morphism is also called an isomorphism.
Observe that relations ( 14)-( 15) in STL imply that the morphisms , , , and are invertible, where .( ) Therefore, (1) ≅ (−1) in the category STL, which implies that the objects in STL are self-dual; that is, every object is isomorphic to its dual.A pivotal category is an autonomous category C where ( * ) * ≅ , for every object  in C. Therefore, categories OTa and STL are examples of pivotal categories.In fact, ( * ) * = , for every object  in OTa.This new category is subject to the same relations as in STL, together with relations (22) depicted below

The Linear Singular
Relations ( 22) "remove" all circles and piecewise oriented circles with two bivalent vertices (whose red triangles point to each other), and replace them with  +  −1 .
Observe that the set of morphisms between objects  and  in LSTL is a Z[,  −1 ] module, which we denote by Hom LSTL (, ), where we mod out by the relations (22) and those displayed in Figure 8, up to isotopy.

The 𝑠𝑙(2) Polynomial Revisited
We define a monoidal functor  : OTa → LSTL from the category of oriented tangles to the linear singular Temperley-Lieb category.The functor  is the identity on objects.Moreover, the functor  is the identity on all of the generating morphisms for OTa, except for and , for which we have

ISRN Geometry
The functor  is well defined, in the sense that all of the defining relations in OTa hold for their images under  in LSTL.The planar relations (a) and (b) follow since these relations hold in both categories, and since  is the identity on the morphisms that are used to form these relations.The image under  of the first equality (c) equals The second equality (c) is verified similarly.For relation (d), observe the following: Therefore, we have which verifies that the first part of relation (d) in OTa holds for its image in LSTL.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 )] −q 2 − q 2 ( q ( q − q 2 ) )] − q 2 − q 2 ( q − q 2 [q(q − q 4 − q 4  + q 5 = q 3 − q 4  + q 5 + q 5 − q 6 − q 4 + q 5 = q 3  − 2q 4   + q −1 (q ) − q 6 + q 5 + q 5 − q 4  − q 4 = q 3 + q 5 + q 5  . ( Applying similar computations to the right hand side of relation (e), we obtain F( − q 4 − q 4 ) = q 3 + q 5  + q 5 , ( And, therefore,  ( ) =  ( ) .For relation (f), we have as desired.Finally, we verify relation (g): Relation (h) is verified similarly, and we leave it for the reader.Therefore, the functor  is well defined.
Our construction implies that if  is an oriented (, ) tangle, then () is an ambient isotopy invariant for .Moreover, an oriented link  is a (0, 0) tangle, and () is an ambient isotopy invariant of  which equals the (2) polynomial of .Thus, our functor  provides another way of defining the (2) link invariant.

Polynomial Invariants for Tangles
The scope of this section is to show how to use the geometric categories STL and LSTL to construct invariants for tangles.
6.1.The Case of (,) Tangle.In this section we restrict our attention to endomorphisms in OTa, STL, and LSTL.Recall that an endomorphism in a category C is morphism  :  →  whose source and target are the same object in C.

The Case of Arbitrary Tangles.
We consider now arbitrary (, ) tangles and construct a polynomial invariant for them.
Given a singular flat tangle , where  :  → , we define the V of  as the singular flat tangle () :  →  obtained by reflecting  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 3 :
Figure 3: Relations in the category OTa.

Figure 6 :
Figure 6: Tensor product of morphisms in STL.
Temperley-Lieb Category.We define the linear singular Temperley-Lieb category and denote it by LSTL, as the free Z[,  −1 ]-linear category generated by STL.The objects of LSTL are the same as those of STL, and the morphisms are Z[,  −1 ]-linear combinations of morphisms in STL.The composition of morphisms in LSTL is defined by bilinear extension of the composition of morphisms in STL.

Example 2 .
Consider the following: