A COMPARISON BETWEEN SLnSPIDER CATEGORIES ANUP POUDEL ABSTRACT . We prove a conjecture of L ˆe and Sikora by providing a comparison between various

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A COMPARISON BETWEEN SLnSPIDER CATEGORIES

ANUP POUDEL

ABSTRACT. We prove a conjecture of Lˆ

e and Sikora by providing a comparison between various

existing SLnskein theories. While doing so, we show that the full subcategory of the spider category,

Sp(SLn), deﬁned by Cautis-Kamnitzer-Morrison, whose objects are monoidally generated by the

standard representation and its dual, is equivalent as a spherical braided category to Sikora’s quotient

category. This also answers a question from Morrison’s Ph.D. thesis. Finally, we show that the skein

modules associated to the CKM and Sikora’s webs are isomorphic.

1. INTRODUCTION

The category of representations of the quantum group Uq(sln)has a spherical and braided tensor

(ribbon) structure. In particular, since it is a pivotal monoidal category one can describe the cate-

gory using diagrammatic calculus. By introducing the notion of combinatorial spiders in [Kup1],

Kuperberg ﬁrst provided a diagrammatic presentation for the category of ﬁnite dimensional repre-

sentations of Uq(g), where gis a simple Lie algebra of rank 2. The diagrammatic presentation for a

representation category has many advantages. For example, diagrammatic presentations lead natu-

rally to the deﬁnition of skein modules. Skein modules (c.f. Def. 7.2) have become central objects

of study in the ﬁeld of quantum topology connecting them to quantum invariants of 3-manifolds,

topological quantum ﬁeld theory, quantum cluster algebras and quantum hyperbolic geometry, see

for example [BW1, BW2, BFK, FG, FKL, Le, Mu, PS] and references within. Using diagrammatic

presentation for a representation category of a quantum group, one obtains a natural description for

its associated skein category (c.f. Section 7) which allows one to understand the associated skein

modules.

Extending Kuperberg’s work, Kim [Kim] proposed a presentation of the category of ﬁnite di-

mensional representations of Uq(sl4)where the colors correspond to the exterior powers of the

standard representation and its dual. Sikora in [Sik] provided a presentation for the braided spher-

ical monoidal category coming from the representation theory of Uq(sln)using the standard rep-

resentation and its dual as objects. Further, Morrison proposed a complete set of generators and

relations (conjecturally) in [Mor] for the spherical monoidal category, Rep(Uq(sln)) where the

colors correspond to the exterior powers of the standard representation and its dual. Later, Cautis,

Kamnitzer and Morrison proved Morrison’s conjecture in [CKM] using skew-Howe duality.

The braided monoidal structure on Uq(sln)was ﬁrst explored diagrammatically by Murakami et

al. in [MOY] (also see [KW]). They provide web relations that align with the untagged relations

(3.5–3.8) in [CKM]. However, they provide no discussion of a complete set of relations for this

category. Later, Sikora [Sik] explained the connection between his presentation for Rep(Uq(sln))

and generators and relations presented in [MOY]. Further, in his thesis [Mor], Morrison poses a

question regarding the relation between his conjecture and the work of Sikora. In this paper, we

2020 Mathematics Subject Classiﬁcation. 57K31, 17B37.

Key words and phrases. Webs, HOMFLYPT relations, skein categories, skein modules.

arXiv:2210.09289v5 [math.GT] 4 Apr 2025

2 ANUP POUDEL

answer Morrison’s question and also prove Conjecture 1.1 [LS] which is related to the question

posed by Morrison in his thesis.

There is a braided spherical category based on the HOMFLYPT skein relations. Early on it

was realized [TW] that by specializing the variables in HOMFLYPT one could obtain a category

that mapped down to the categories of Uq(sln)representations. One can build skeins that behave

algebraically like Young symmetrizers [Y, M, MA, Li, B]. The category is missing both generators

and relations that say that the nth exterior power of the standard representation and its dual are

trivial. Sikora’s model adds n-valent vertices that are sources and sinks corresponding to these

invariant tensors and a relation for cancelling them. The CKM model adds tags that are sources

and sinks and relations for moving them and cancelling them. The work in this paper shows that

the two approaches are equivalent.

As in [LS], let Sb

nbe a monoidal category with ﬁnite sequences of signs ±as objects and isotopy

classes of n−tangles (cf. [Sik]) as morphisms. The tensor product is given by horizontal concate-

nation and composition of morphisms is given by vertical stacking. The category of modules in

[LS] are over a commutative ring with a distinguished invertible element.

Let Cnbe the category of left Uq(sln)-modules isomorphic to ﬁnite tensor products of Vand

V∗where Vis the deﬁning representation of Uq(sln). Deﬁne a monoidal functor RT0:Sb

n→Cn

which for any object η={η1,···ηk} ∈ Sb

n, is deﬁned as RT0(η):=Vη=Vη1⊗ ·· · ⊗Vηk. Note

that V+=Vand V−=V∗. For any n−tangle, the functor takes caps and cups to evaluation and

coevaluation maps respectively, crossings to the braid isomorphisms and an n−sink (resp. source)

to a map from the n−fold tensor of V(resp. ground ring) to the ground ring (resp. n−fold tensor

of V). Also, a monoidal ideal in a monoidal category, Cis a subset I⊂Hom(C)such that for

x∈Iand y∈Hom(C), we have x⊗y,y⊗x∈Iand x◦y,y◦x∈Iwhenever such compositions are

deﬁned.

Conjecture 1.1 ([LS]).The kernel kerRT0is the monoidal ideal generated by elements given in

relations (3.11–3.14).

In this paper, we prove the Conjecture 1.1, over an integral domain Rwhere certain quantized

integers are invertible (c.f. Section 2), by proving that Sikora’s braided spherical category is equiv-

alent (as a braided spherical category) to the full subcategory of the braided spherical category

Sp(SLn)in [CKM] which has as objects the standard representation and its dual.

1.1. Main results: The main results of this paper are:

•Theorem 6.5 which shows that the braided spherical category coming from [Sik] and the

full subcategory of the spider category in [CKM] with the standard representation and its

dual as objects are equivalent to each other.

•Theorem 6.6 which provides a proof for the Conjecture 1.1 under our choice of integral

domain.

•Theorem 7.2 which shows that the skein modules associated to the Sikora webs is isomor-

phic to those associated to the CKM webs.

1.2. Outline. In Section 2, we deﬁne the quantized integers (and binomial coefﬁcients) along

with the categorical structures that appear in our work. The notion of a free spider category and

operations in this category are also introduced. We recall the deﬁnitions of the two main categories

in this paper: the CKM spider category and Sikora’s spherical braided category in Section 3. We

also recall the deﬁnition of the MOY category which is closely related to the CKM category.

In Section 4, the CKM box relations are derived in a diagrammatic fashion using the braided

A COMPARISON BETWEEN SLnSPIDER CATEGORIES 3

structure of the CKM spider category showing that some of the CKM relations in the braided

setting are redundant. This allows us to relate the CKM and Sikora categories easily later in the

paper. In Section 5, we introduce and work with subcategories of (the full subcategory) the MOY

category to show that relations in the MOY category is completely characterized by the specialized

HOMFLYPT relation (c.f. Figure 2) in Theorem 5.2. This serves as the ﬁrst step toward relating

the CKM and Sikora categories. In Section 6, with the aid of the main theorem from the previous

section and the antisymmetrizer relation (c.f. 6.3) we prove that Sikora’s spherical braided category

is equivalent (as a spherical braided category) to the full subcategory of the CKM spider category.

Further, using this result, we prove the Conjecture 1.1 of Lˆ

e and Sikora. In Section 7, we extend

our result regarding the ribbon categories to an equivalence of skein categories. As a consequence,

we we show that the skein module associated to the CKM spider category is isomorphic to the one

associated to Sikora’s spider category (Theorem 7.2).

1.3. Acknowledgements: Part of this work was done during the author’s PhD dissertation work.

The author is grateful to his PhD advisor, Charles Frohman for many helpful discussions and guid-

ance. The author would also like to thank Thang Lˆ

e for helpful discussions, and the anonymous

referees for many useful suggestions and comments on the earlier version of the paper.

2. PRELIMINARIES

2.1. Coefﬁcients. Let Rbe an integral domain containing 1, and suppose that q∈Ris a unit. The

quantized integers in Rare deﬁned to be the sums

[k] =

k−1

∑

i=0

q−k+1+2i.(2.1)

The quantized factorials are deﬁned recursively by [0]!=1 and [n]!= [n][n−1]!, and the quantum

binomial coefﬁcients n

k=[n]!

[k]![n−k]!.(2.2)

We will assume that we are working over a ring Rhaving a unit q1/nso that if the category is

associated with slnthen the quantum integers [1],...,[n]are also units.

2.2. Categories. Apivotal monoidal category,C, is a rigid monoidal category such that there

exist a collection of isomorphisms (a pivotal structure) aX:X∼

−→ X∗∗, natural in X, and satisfying

aX⊗Y=aX⊗aYfor all objects X,Yin C.

For a pivotal monoidal category, C, an object X∈C, and morphisms f∈End(X), we deﬁne

the left and right quantum traces,Trl(f)and Trr(f)respectively as follows (see [Tur1] for more

details):

Trl(f) = evX◦(idX∗⊗f)◦coev′

X∈End(1)(2.3a)

Trr(f) = ev′

X◦(f⊗idX∗)◦coevX∈End(1)(2.3b)

where (co)evXand (co)ev′

Xare left and right (co)evaluations, respectively deﬁned as:

evX:X∗⊗X→1coevX:1→X⊗X∗

ev′

X:X⊗∗X→1coev′

X:1→∗X⊗X

Further, Cis a spherical monoidal category if it is a pivotal category such that the left and right

quantum traces are the same. In a spherical monoidal category, the quantum dimension dXof an

object Xis deﬁned to be the quantum trace of identity, idX. Further, note that dX=dX∗.

4 ANUP POUDEL

Abraided monoidal category Cis a monoidal category such that there exist a collection of natu-

ral isomorphisms (braid isomorphisms)βX,Y:X⊗Y→Y⊗Xfor any pair of objects X,Y∈Cthat

are compatible with the associativity isomorphisms. This compatibility with the associativity iso-

morphisms in the monoidal category is ensured by the hexagon axiom that the braid isomorphisms

satisfy. We refer the reader to [Tur1] for more details.

We work with spherical braided (ribbon) categories via generators and relations. The generators

are diagrams carrying labels, where the labels represent irreducible modules over some semisimple

Lie algebra. The diagrams represent an element (a vector) in the morphism space (a vector space)

of the corresponding category of representations of the Lie algebra. Further, each diagram is con-

sidered up to regular isotopy. In the absence of relations this is called the free spider category (on

whatever the generators are). The operations are given by (as deﬁned in [Kup1]) the following:

Join: This operation simply allows one to tensor two diagrams (morphisms) by horizontally con-

catenating.

Stitch: For any diagram in Hom(A,B), this means composing with an evaluation (or coeval-

uation) to attach a cap or a cup. So, as an example, a stitch could send Hom(A⊗B,C)to

Hom(A⊗B⊗B∗,C)∼

=Hom(A,C).

Rotation: This allows one to apply a cyclic permutation on the tensor factors (up to sign). Dia-

grammatically, this amounts to attaching a cap and a cup to rotate the diagram.

Stitch Rotate

Note that given a spherical tensor category, these (diagrammatic) operations already exist coming

from the morphisms in the category.

3. THE BRAIDED CATEGORIES

3.1. The CKM braided category. As in [CKM], let Rep(SLn)be the category of Uq(sln)-modules

generated by tensor products of the fundamental representations. This is a braided spherical

monoidal category which is a full subcategory of the category of representations of Uq(sln)where

all the morphisms are generated by the wedge product and a version of its adjoint that embeds

Λk+lCninto ΛkCn⊗ΛlCn:

ΛkCn⊗ΛlCn→Λk+lCnand Λk+lCn→ΛkCn⊗ΛlCn(3.1)

The free spider category FSp(SLn): Deﬁne the free spider category to be freely generated by the

planar diagrams for morphisms in Rep(SLn)as shown below. Objects in FSp(SLn)are subse-

quences of {1±,...,(n−1)±}, where ‘+’ denotes an arrow going upward and ‘-’ denotes an arrow

pointing downward. The morphisms are generated by the following:

FIGURE 1. Generators for FSp(SLn)

A COMPARISON BETWEEN SLnSPIDER CATEGORIES 5

The spider category, Sp(SLn), is the quotient of FSp(SLn)by the following relations (together

with their mirror reﬂections and arrow reversals) [CKM]:

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

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ACOMPARISONBETWEENSLnSPIDERCATEGORIESANUPPOUDELABSTRACT.WeproveaconjectureofLˆeandSikorabyprovidingacomparisonbetweenvariousexistingSLnskeintheories.Whiledoingso,weshowthatthefullsubcategoryofthespidercategory,Sp(SLn),definedbyCautis-Kamnitzer-Morrison,whoseobjectsaremonoidallygeneratedbythestandard...

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A COMPARISON BETWEEN SLnSPIDER CATEGORIES ANUP POUDEL ABSTRACT . We prove a conjecture of L ˆe and Sikora by providing a comparison between various

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