The skein algebra of the Borromean rings complement

The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.

1. Introduction 1.1. Classical limit of Kauffman bracket skein module and skein algebra. Let M be an an oriented 3-manifold and R a commutative ring with unit 1. Przytycki [10,11] and Turaev [14,15] introduced the Kauffman bracket skein module (KBSM) S(M ; R, t) as a generalization of the Kauffman bracket in S 3 . The KBSM of M is defined by the Kauffman bracket skein relation − t − t −1 and the weight relation + (t 2 + t −2 )∅ dividing the free R-module spanned by framed links in M . For link complements in S 3 , the KBSM of two-bridge links are calculated [2,3,7,8]. For other links and general parameter t, the KBSM is less researched because its algebraic structure is too complicated.
The classical case t = −1 of the KBSM is relatively simple to be calculated and is important; in this case the framing becomes irrelevant and S(M ; R, −1) admits a commutative algebra structure with the multiplication as the disjoint union of links. Bullock [1] showed that the algebra S(M ; C, −1) modulo its nilradical is isomorphic to the coordinate ring of the SL 2 (C)-character variety of M . From this point of view the KBSM can be seen as a deformation of SL 2 (C)-character variety, and plays an important role in the study of the AJ conjecture [5,7,9], which relates the recurrence polynomial of the colored Jones polynomial and the A-polynomial.
In [12,13], Przytycki and Sikora introduced the skein algebra S(G; R) of a group G, which is a generalization of S(M ; R, −1) in the sense that S(M ; R, −1) can be recovered when G = π 1 (M ). They also showed that the skein algebra S(G; C) modulo its nilradical is isomorphic to the coordinate ring of the SL 2 (C)-character variety of G.
For link complements in S 3 , Tran [16,17] calculated the skein algebras of pretzel links. Note that the fundamental groups of the complements of pretzel links and two-bridge links are generated by two elements. In this paper we determine the skein algebra of a group with three generators and two relators, and give an explicit formula for the skein algebra, i.e., the KBSM with t = −1, of the Borromean rings complement in S 3 . Horowitz [6], Culler and Shalen [4] showed that if G is generated by g 1 , g 2 , . . . , g n , then each SL 2 (C)-character χ is determined by the values χ(g i1 ) · · · χ(g i k ), 1 ≤ k ≤ n, 1 ≤ i 1 < i 2 < · · · < i k ≤ n. This result implies that the skein algebra S(G; C) is generated by {[g i1 · · · g i k ] | 1 ≤ k ≤ n, 1 ≤ i 1 < · · · < i k ≤ n}. The main interest of the present paper is the relations among these generators.
Let F n be the free group with generators g 1 , . . . , g n . As for the free groups F 1 and F 2 , there are no relations among the above generators of S(F 1 ; C) and S(F 2 ; C), and therefore S(F 1 ; C) and S(F 2 ; C) are nothing but the polynomial rings , respectively. However, as for F n with n larger than 2, the skein algebra is no longer a polynomial ring of the variables {[g i1 · · · g i k ] | 1 ≤ k ≤ n, 1 ≤ i 1 < · · · < i k ≤ n}, but is the quotient of the polynomial ring by a certain ideal. In order to describe this ideal we consider the algebra homomorphism from the polynomial ring C[x 1 , . . . , x 1···n ] with 2 n − 1 variables x i1···i k (1 ≤ k ≤ n, 1 ≤ i 1 < i 2 < · · · < i k ≤ n) to the skein algebra S(F n ; C) as follows: Since Φ is surjective, S(F n ; C) is isomorphic to C[x 1 , . . . , x 1···n ]/ ker Φ. For n = 3, by the argument in [6], we have ker Φ = K , where i.e., we have Let G be a group with generators g 1 , . . . , g n and set ι : F n ∋ g →ḡ ∈ G the natural group homomorphism such that ι(g i ) = g i for 1 ≤ i ≤ n. We extend ι to the algebra homomorphism ι : S(F n ; C) → S(G; C) and define Φ G := ι • Φ. Then we have C[x 1 , . . . , x 1···n ]/ ker Φ G ∼ = S(G; C).
For n = 1 or 2, it is known that where P g is any polynomial such that Φ(P g ) = [g].
We determine ker Φ G for n = 3 as follows.
Theorem 1. Let G be a group with three generators. Then we have Note that I G is generated by a finite set of polynomials, since the polynomial ring is Noetherian. The following theorem gives a finite set of generators of I G when G has two relators.
Theorem 2. Let G be a group defined by three generators g 1 , g 2 , g 3 and two relators α = β and γ = δ. Then I G is generated by By the Wirtinger presentation of the Borromean rings B, we have and δ = g −1 1 g 3 g 1 g −1 3 g 2 (Lemma 10). Using Theorem 2 we give explicit generators of I π1MB as follows.
Theorem 3. For the Borromean rings complement M B , let I π1MB be the ideal generated by the polynomials below. Then we have S(π 1 M B ; C) ∼ = C[x 1 , . . . , x 123 ]/I π1MB .
We can find some symmetries in the polynomials above. For example, if we multiply Q αg1,βg1 by −1 and exchange each "2" and "3" in the indices of the variables except for x 23 and x 123 , then we obtain Q γg1,δg1 . This kind of symmetry can be observed between Q αg1,βg1 and Q γg1,δg1 , Q αg2,βg2 and Q γg3,δg3 and among Q γg1g2,δg1g2 , Q αg2g3,βg2g3 and Q γg2g3,δg2g3 . Each Q αg1g2,βg1g2 , Q γg1g3,δg1g3 and Q γg1g2g3,δg1g2g3 has a symmetry by themselves.
Organization of this paper. The rest of the paper is organized as follows. In Section 2 we study the skein algebra of groups with three generators and two relators, and prove Theorem 1 and Theorem 2. In Section 3 we consider the skein algebra of the complement M B of the Borromean rings B. In Section 3.1 using Wirtinger's method we give a presentation of π 1 M B with three generators and two relators, and in Section 3.2 we prove Theorem 3.

Skein algebra of groups with three generators and two relators
In this section we prove Theorem 1 and Theorem 2.
2.1. Proof of Theorem 1. We prove Theorem 1. Let G be a group with three generators g 1 , g 2 , g 3 . Recall from the introduction the natural group homomorphism ι : F 3 ∋ g →ḡ ∈ G such that ι(g i ) = g i for i = 1, 2, 3. We use the following lemma.
Proof. Let us consider the following two ideals of the symmetric algebra SC[F 3 ]: If we consider the canonical projection then we haveĪ Thus by the third isomorphism theorem, we have Hence it suffices to show that SC[F 3 ]/Ī ∼ = S(G; C).
We extend the group homomorphism ι : F 3 → G to the algebra homomorphism ι : . Then ι induces the algebra isomorphismι : , where the kernel ker ι is the ideal generated by u − v, u, v ∈ F 3 , such that u =v. Then again by the third isomorphism theorem, we have where the last identity is given bŷ which is the defining relation of the skein algebra S(G; C).
Proof of Theorem 1. Recall from the introduction that the map Φ : C[x 1 , . . . , x 123 ] → S(F 3 ; C) has the kernel generated by the polynomial K, i,e., we have the isomor-phismΦ Then by the third isomorphism theorem, we have By Lemma 4 the right hand side is isomorphic to S(G; C), which implies the assertion.
Lemma 5. Let G be a group and a, b, c ∈ G arbitrary elements. In S(G; R), we have the following identities.
[a] = [a −1 ] (1) 2.3. Proof of Theorem 2. We prove Theorem 2. Let and consider the ideal Since generators ofĪ G are all contained in I G , it suffices to show thatĪ G ⊃ I G . Since K ∈Ī G , it is enough to prove the following lemma.
We reduce the assertion to the following claim: For P u , P v , P a1c1 and P bmcm we have We first prove (a) and (c). Note that Thus we have In order to prove (b), we use the following claim.
By combining (b1) and (b2) we obtain (b); As for (b1), the proof is given under more general condition in Lemma 9 below. We prove (b2).
• subcase 2-2. m 1 > 1 We use induction on m 1 . Set where the assertion for m 1 = 2 follows from the case 1 and the subcase 2-1, that for m 1 = 3 follows from the subcase 2-1 and the case m 1 = 2, and that for m 1 ≥ 4 follows from the induction on m 1 .
and the proof is similar to the subcase 2-2.
In this case by the definition ofĪ G we have P αgi 1 gi 2 − P βgi 1 gi 2 ∈Ī G .
• subcase 3-2. m 1 = 1, which follows from (4) in Lemma 5. Then we have We use induction on m 2 . Set where the assertion for m 2 = 2 follows from the case 2 and subcases 3-1 and 3-2, that for m 2 = 3 follows from the subcases 3-1 and 3-2 and the case m 2 = 2, and that for m 2 ≥ 4 follows from the induction on m 2 .
We use induction on m 1 . Set , which follows from (3) in Lemma 5. Then we have where the assertion for m 1 = 2 follows from case 2 and subcases 3-1 through 3-4, that for m 1 = 3 follows from subcases 3-1 through 3-4 and the case m 1 = 2, and that for m 1 ≥ 4 follows from the induction on m 1 .
We use induction on |m 1 |. Set and the proof is similar to the subcase 3-5.
For (i 1 , i 2 , i 3 ) = (2, 3, 1), we define We use induction on m 3 . Set , which follows from the defining relation gh = g ⊗ h − gh −1 in the skein algebra. Then we have where the assertion for m 3 = 2 follows from the case 3 and the subcase 4-1, that for m 3 = 3 follows from the subcase 4-1 and the case m 3 = 2, and that for m 3 ≥ 4 follows from the induction on m 3 .
We use induction on m 2 . Set , which follows from (3) in Lemma 5. Then we have where the assertion for m 2 = 2 follows from the case 3 and the subcases 4-1 through 4-3, that for m 2 = 3 follows from the subcases 4-1 through 4-3 and the case m 2 = 2, and that for m 2 ≥ 4 follows from the induction on m 2 .
We use induction on |m 2 |. Set and the proof is similar to the subcase 4-4.
We use induction on m 1 . Set , which follows from (3) in Lemma 5. Then we have where the assertion for m 1 = 2 follows from the case 3 and subcases 4-1 through 4-5, that for m 1 = 3 follows from the subcases 4-1 through 4-5 and the case m 1 = 2, and that for m 1 ≥ 4 follows from the induction on m 1 .
We use induction on |m 1 |. Set and the proof is similar to the subcase 4-6.

Skein algebra of Borromean rings complement in S 3
In this section we consider the skein algebra of the fundamental group π 1 M B of the Borromean rings complement M B . We give explicit generators of I π1MB .
3.1. Fundamental group of Borromean rings complement. Let M B be the Borromean rings complement in S 3 . We will use the following presentation of π 1 M B .
3.2. Proof of Theorem 3. In this subsection we prove Theorem 3. Recall the polynomials Q αg,βg , Q γg,δg in Theorem 3. We define Q α,β , Q γ,δ , Q γg2,δg2 , Q αg3,βg3 , which are not on the list in Theorem 3, as copies of the zero polynomial, and consider the ideal

Recall that we have
by Theorem 2, and thus it suffices to show thatĪ π1MB =Î π1MB .
We reduce Theorem 3 to the following lemma.
Proof of Theorem 3 assuming Lemma 11. By Lemma 11 we haveĪ π1MB =Î π1MB because This completes the proof.
In what follows, we prove Lemma 11. We will use the following formulae.
Lemma 12. We have Proof. (a1) and (a2) follow from the defining relations and [e] = 2 of the skein algebra. For (b1), using Lemma 5 (4), we have We can prove (b2) and (b3) similarly.  Q αg1,βg1 and Q γg1,δg1 We have Here we have Using the above identities and Lemma 12 (b), (c1), we have On the other hand, we have Here we have Using the above identities and Lemma 12 (b), (c1), by a straight calculation we have = Φ(Q γg1,δg1 ).