The Hilb-vs-Quot conjecture

In this appendix, we specify our grading conventions for Khovanov–Rozansky homology, compare them to those of other published works, and illustrate on the smallest examples (unknot, Hopf link, trefoil, ( 3 , 4 ) torus knot) to aid the reader’s sanity. Our exposition closely follows [15, §1.6], but we correct some mistakes: see Remarks A.1–A.2.

A.2 Soergel bimodules

Let T = G m n and S : = H T ∗ ( pt ) = C [ t 1 , … , t n ] . We regard 𝐒 as a graded ring, with deg ⁡ ( t i ) = 2 for all 𝑖. Thus the S n -action on 𝑇 that permutes coordinates also preserves the grading on 𝐒. Let s i ∈ S n be the transposition that swaps t i and t i + 1 .

In the category of graded 𝐒-bimodules, we write ( m ) for the grading shift B ⁢ ( m ) i = B i + m . Let S ⁢ Bim be the full subcategory generated by the identity bimodule 𝐒 and the bimodules S ⊗ S s i S ⁢ ( 1 ) for all 𝑖 under isomorphisms, direct sums, tensor products ⊗ ⁣ = ⁣ ⊗ S , direct summands, and grading shifts. Objects of S ⁢ Bim are called Soergel bimodules. We write K b ⁢ ( S ⁢ Bim ) for the bounded homotopy category, a monoidal additive category under ⊗.

Let Br n be the group of braids on 𝑛 strands up to isotopy. Any braid β ∈ Br n defines an object T ̄ β ∈ K b ⁢ ( S ⁢ Bim ) called the Rouquier complex of 𝛽. See, e.g., [15, §2.1] for the precise definition.

Let Vect 2 be the category of Z 2 -graded vector spaces that are finite-dimensional in each bidegree, such that the first grading is bounded below and the second is bounded. Let

be the Hochschild cohomology functor

These Ext’s can be computed using a Koszul resolution of 𝐒 over S ⊗ C S op , which shows that the Ext grading sits in degrees 0 through (at most) 𝑛.

Let Vect 3 be the category of Z 3 -graded vector spaces that are finite-dimensional in each tridegree, such that the first grading is bounded below and the other two gradings are bounded. Let HHH ̄ = HHH ̄ ∗ , ∗ , ∗ be the composition of functors

Explicitly, the gradings are ordered so that

The story above can be redone with the quotient torus T 0 : = T / T S n in place of 𝑇. Note that T 0 is just the image of 𝑇 along the quotient map GL n → PGL n . Replacing 𝑇 with T 0 entails replacing 𝐒 with its subring S 0 : = H T 0 ∗ ( pt ) . We write T β , HH , HHH for the objects that respectively replace T ̄ β , HH ̄ , HHH ̄ .

Let 𝐿 be the link closure of 𝛽. In [34], Khovanov proved that HHH ⁢ ( T β ) matches the reduced version of the triply graded homology of 𝐿 proposed in [10] and constructed in [35], up to an affine transformation of the trigrading. One can show that

and that, in consequence, HHH ̄ ⁢ ( T ̄ β ) matches the unreduced version of the homology constructed in [35], up to similar regradings.

A.3 The main dictionary

For any β ∈ Br n , let

That is,

1. hhh ̄ β ⁢ ( A , Q , T ) is the series denoted P β ⁢ ( Q , A , T ) in [11, §A] and [15, §1.6], and hhh β is the analogue of hhh ̄ for reduced homology.

1. P ̄ L norm ⁢ ( A , Q , T ) for the series denoted P L norm ⁢ ( Q , A , T ) in [15],

2. P L , ORS ⁢ ( a , q , t ) for the series denoted P ⁢ ( L ) in [42] (it is denoted P ⁢ ( L − ) in [10], where L − is the chiral mirror of 𝐿),

3. P ̄ L , ORS ⁢ ( a , q , t ) for the series denoted P ̄ ⁢ ( L ) in [42], which satisfies

   (A.2) P ̄ L , ORS ⁢ ( a , q , t ) = P ̄ U , ORS ⁢ ( a , q , t ) ⁢ P L , ORS ⁢ ( a , q , t ) .

Let 𝑒 be the writhe of 𝛽, meaning its net number of crossings counted with sign, and let 𝑏 be the number of components of 𝐿. After correction, [15, §1.6] states

In general, we will not work with P ̄ L norm . Moreover, we will not discuss at all the normalizations used in the series P ⁢ ( U ) , P ⁢ ( T ⁢ ( 2 , 3 ) ) in [15, Remark 1.27].

A.4 “Our” series

For any braid β ∈ Br n with writhe 𝑒 whose link closure 𝐿 has 𝑏 components, let

Above, note that X id ⁢ ( a , q , t ) = 1 + a 1 − q . We can check that

It turns out that, in the rest of this paper, X ̄ β and X β are the most convenient series for us to use.

In particular, suppose that f ( x , y ) ∈ C ⟦ x ⟧ [ y ] such that f ⁢ ( x , y ) = 0 defines a generically separable, degree-𝑛 cover of the 𝑥-axis, fully ramified at ( x , y ) = ( 0 , 0 ) . Then the preimage in the cover of a positively oriented loop around x = 0 is a braid β f ∈ Br n , whose link closure is the link L f introduced in Section 1.4. We see that X ̄ β f is precisely the series X ̄ f introduced in (1.4).

A.5 Torus links

For integers n , d > 0 , let T n , d be the positive ( n , d ) torus link, considered negative in [10]. Its number of components is b = gcd ⁡ ( n , d ) . Taking f ⁢ ( x , y ) = y n − x d in the construction above shows that T n , d is the link closure of a braid β n , d ∈ Br n for which e = ( n − 1 ) ⁢ d . Let

Let X ̄ n , d = X ̄ β n , d , as in the rest of this paper, and X n , d = X β n , d .

In Section 4, we implicitly need the following identities that match X ̄ n , d , X n , d with other series in the literature.

1. Let P ̃ n , m ⁢ ( u , q , t ) be the series in [23]. For coprime n , d , we have

   X ̄ n , d ⁢ ( a , q , t ) = t δ 1 − q ⁢ P ̃ n , d ⁢ ( − a , q , t − 1 ) .

2. Let P m , n = P m , n ⁢ ( a , q , t ) be the series in [41]. For coprime n , d , we have

   X ̄ n , d ⁢ ( a , q , t ) = ( − a − 1 ⁢ q 1 2 ⁢ t 1 2 ) δ ⁢ P n , d ⁢ ( − a , q , t − 1 ) .

   Note that the substitution sends t ↦ q and q ↦ t − 1 , not vice versa.

3. Let P ̂ 0 M , 0 N ⁢ ( q , t , a ) , Q ̂ 0 M , 0 N ⁢ ( q , t , a ) , R 0 M , 0 N ⁢ ( q , t , a ) be the series in [22]. For any n , d , we have

   1 1 + a ⁢ X ̄ n , d ⁢ ( a , q , t ) = 1 1 − q ⁢ X n , d ⁢ ( a , q , t − 1 ) = R 0 n , 0 d ⁢ ( q , t − 1 , aq − 1 ) = Q ̂ 0 n , 0 d ⁢ ( q , t − 1 , aq − 1 ) by ⁢ [22, Corollary 5.10] = q − d − n ⁢ P ̂ 0 n , 0 d ⁢ ( q , t , aq − 1 ) by ⁢ [22, (11)] .