Research Article | Open Access
Carmen Caprau, Joel Smith, "The Singular Temperley-Lieb Category", International Scholarly Research Notices, vol. 2014, Article ID 321509, 9 pages, 2014. https://doi.org/10.1155/2014/321509
The Singular Temperley-Lieb Category
We introduce and study the singular Temperley-Lieb category over , 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 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.
The Temperley-Lieb algebra has played a central role in the discovery of the Jones polynomial  and in the subsequent developments relating to knot theory, topological quantum field theory, and statistical mechanics . 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 . In , 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  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 , and independently by the first author  that there is a way to resolve the sign indeterminacy in the functoriality of the Khovanov homology. In , 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 link invariant, which is a state summation model for the unnormalized Jones polynomial (with ) via webs, which are oriented bivalent graphs whose vertices are either “sinks” or “sources.”
This state model for the 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 , which we refer to as the singular Temperley-Lieb category. It turns out that is an autonomous category in which the objects are self-dual; that is, each object is isomorphic to its dual. The link polynomial can be interpreted via representation theory of the quantum group . Specifically, an oriented edge in a singular flat tangle stands for the standard (2-dimensional) vector representation of quantum . Since is self-dual (i.e., isomorphic to its dual representation ), it is not surprising that every object in the singular Temperley-Lieb category 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  to singular flat tangles and construct an tangle (co)homology, which should be isomorphic to that developed in .
The paper is organized as follows. We review the polynomial in Section 2. Following Turaev , in Section 3 we briefly review the category of oriented tangles 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 , where is an extension of by allowing formal linear combinations of singular flat tangles with coefficients in . The functor mimics the skein relations defining the 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 by regarding a singular flat tangle as a directed ribbon graph.
2. A State Model for the Polynomial
The polynomial can be defined as follows. Let be a planar projection of an oriented link and decompose each crossing of as explained below (1)
We define , the bracket of , as the linear combination of the brackets of all resolutions of , where is evaluated according the following rules: (2)
Specifically, , where the sum is over all resolutions (states) of , and the signs and are determined by the decomposition rules given in (1).
It is easy to see that . It is known that whenever and are ambient isotopic. This together with the above skein relations implies that is the link invariant or, equivalently, the unnormalized Jones polynomial of the oriented link , with .
A resolution associated with the link diagram is an web, which is an oriented bivalent planar graph whose vertices are either “sources” or “sinks.” A resolution might have a component with no vertices, that is, a closed oriented loop. For each vertex of an 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 tangle is an embedding of a finite collection of arcs (homeomorphic to the interval ) and circles in such that the target endpoints of lie on and the source endpoints of lie on . Knots and links are tangles and braids with strands are the most well-known class of tangles. The number of arcs in an tangle is ; thus, is required to be even.
A diagram of an tangle is a regular projection of the tangle into . Two tangles are (or ambient isotopic) if one can be transformed into the other by an isotopy of which fixes the boundary.
We will work with oriented tangles, that is, tangles whose components are oriented. If is an oriented tangle, then the orientation of an arc in induces orientations of its endpoints as follows: if an arc is oriented up (resp., down) near an endpoint of , then the endpoint receives the positive orientation (resp., the negative orientation ). Thus, to each oriented tangle we associate two sequences of numbers, and , denoted and , respectively, where and . If (resp., ), then (resp., ) is the empty sequence . In Figure 1 we provide an example of a tangle diagram with and .
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 .
We will denote the category of oriented tangles as . The objects of this category are finite sequences of ’s and −1’s, together with the empty sequence . A from to is an isotopy type of an oriented tangle with and . The category 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 .
Turaev  showed that the morphisms in are generated under composition and tensor product by the following morphisms: (4) subject to the relations – 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 the polynomial described in Section 2, we define the singular Temperley-Lieb category, which we shall denote by . The objects in are the same as those in , that is, finite sequences of ’s and ’s, together with the empty sequence (which is the identity object in ). A morphism between objects and is an 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 for labeling the boundary points of a singular flat tangle. An example of a -singular flat tangle with source and target 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 .
4.1. Generators and Relations
It is easy to see that the singular Temperley-Lieb category 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)
We remark that the set of morphisms between objects and in the category is a module, where we mod out by the relations displayed in Figure 8, up to isotopy. We denote this space by .
4.2. 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 . For that, we need to recall some concepts from category theory.
Let be a monoidal category with identity object . An object in has a dual, denoted by , if there exists a pair of morphisms and , such that
If every object in has a dual, then 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 of oriented tangles is an autonomous category, where and . To see this, let and , and observe that (19) are equivalent to (16) for the object in . Similarly, let and , and recall that the following holds in (20) which are pictorial representations for (16) corresponding to the object .
Since the above relations hold in the category as well, the following statement holds.
Proposition 1. The singular Temperley-Lieb category is an autonomous category.
Relations - in and equivalently relations (12)-(13) in imply that and are dual morphisms (of each other) in both categories, and . Moreover, in 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 is said to be if there exists a morphism in such that and . In this case, is said to be the inverse of and is denoted by . If a morphism is invertible, then is said to be to and denoted by . An invertible morphism is also called an isomorphism.
Observe that relations (14)-(15) in imply that the morphisms , , , and are invertible, where (21) Therefore, in the category , which implies that the objects in are self-dual; that is, every object is isomorphic to its dual.
A pivotal category is an autonomous category where , for every object in . Therefore, categories and are examples of pivotal categories. In fact, , for every object in .
4.3. The Linear Singular Temperley-Lieb Category
We define the linear singular Temperley-Lieb category and denote it by , as the free -linear category generated by . The objects of are the same as those of , and the morphisms are -linear combinations of morphisms in . The composition of morphisms in is defined by bilinear extension of the composition of morphisms in .
This new category is subject to the same relations as in , 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 .
5. The Polynomial Revisited
We define a monoidal functor 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 , except for and , for which we have (23)
The functor is well defined, in the sense that all of the defining relations in hold for their images under in . The planar relations and 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 equals (24) The second equality is verified similarly. For relation , observe the following: (25)
Therefore, we have (26) which verifies that the first part of relation in holds for its image in . The second relation is verified in a similar manner.
Now we take a look at relation . The image of the left hand side of the equality equals(27)
Applying similar computations to the right hand side of relation , we obtain (28) And, therefore, = .
For relation , we have (29) as desired. Finally, we verify relation : (30) Relation 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 tangle, and is an ambient isotopy invariant of which equals the polynomial of . Thus, our functor provides another way of defining the link invariant.
6. Polynomial Invariants for Tangles
The scope of this section is to show how to use the geometric categories and to construct invariants for tangles.
6.1. The Case of Tangle
In this section we restrict our attention to endomorphisms in , and . Recall that an endomorphism in a category is morphism whose source and target are the same object in .
The trace of an endomorphism in a pivotal category with identity object is the endomorphism of given by the formula
For any endomorphism in obtained by applying the functor to a tangle in , its trace, , is a polynomial in . We regard as a map on the space of endomorphisms , and extend it by bilinearity to .
Example 2. Consider the following: (32)
Definition 3. Given an tangle , define a polynomial by
Proposition 4. If is a diagram of an oriented tangle , then is an ambient isotopy invariant for .
Proof. Since the functor is well defined, whenever and differ by a Reidemeister move. In this case, , and therefore, is independent of the diagram .
Example 5. Consider the oriented tangle diagram . Applying the functor , we have the following linear combination of singular flat tangles: (34) Applying the trace function, we obtain (35) We have obtained that () = .
6.2. 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 of as the singular flat tangle obtained by reflecting about the line , 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.
We regard as a map on the space and extend it by linearity to .
Definition 6. Given a singular flat tangle , define .
Observe that and that it can be extended to by bilinearity. Moreover, is an endomorphism of in , and thus we can take the trace of it, .
Definition 7. Given an oriented tangle , define the Laurent polynomial by
The following statement follows at once from our construction.
Proposition 8. is an ambient isotopy invariant for oriented tangles.
Example 9. Find the polynomial for (38) Applying the functor to , we have (39) The involution tangles of the singular flat tangles found in the previous step are (40) Composing all singular flat tangles obtained in the first step with their corresponding involution tangle , we obtain (41) Finally, applying the trace function and the relations (22), we have (42) Therefore, .
7. Ribbon Graphs and Singular Flat Tangles
A band is the image of the square under its embedding in , and an annulus is the image of the cylinder under its embedding in . The core of the band (equivalently the core of the annulus) is the image of the segment (equivalently ) under this embedding.
Let and be nonnegative integers with even. A ribbon graph is an oriented surface embedded in and decomposed into bands and a finite collection of annuli. The choice of orientation for is equivalent to a choice of one side of . Rotating a band (or an annulus) by around its core results in the same band (or annulus) with the opposite orientation.
A ribbon graph is called directed if the cores of its bands and annuli are equipped with directions. Note that the cores of the bands and annuli of the surface are provided with opposite directions on the two sides of .
With each directed ribbon graph we associate two sequences and of in a similar fashion as we did for oriented tangles and singular flat tangles.
We can regard a singular flat tangle as a directed ribbon graph . Since a bivalent vertex in is a singularity where orientations disagree, it corresponds to a half-twist in a band or an annulus of the ribbon graph . If the red triangle corresponding to a bivalent vertex in 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 Section 4.2, the above generators for STL are isomorphisms, and thus . It implies that the category is self-dual, where there exists an isomorphism for every object (represented as a half-twist).
For let , and for let = . Observe that if = , then its inverse is = , and their composition is (44)
We remark that the self-duality imposes a braiding on STL via the isomorphisms given by for any pair of objects and .
Moreover, for every object in , there is an isomorphism given by and represented as a full twist on . For example, if , then and is represented as (46) Note that and for any objects in , which implies that the category is tortile (see ).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
During the research of this paper, the second author was partially supported by the California State University, Fresno, CA, USA, through an undergraduate research grant.
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Copyright © 2014 Carmen Caprau and Joel Smith. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.