Skein modules were introduced by the second author in [Prz1] with the goal of building an algebraic topology based on knots. ey generalise the skein theory of the various link polynomials in S 3 , for example, the Alexander, Jones, Kau man bracket, and HOMFLYPT polynomial link invariants, to arbitrary 3-manifolds. e skein module based on the Kau man bracket skein relation is the most comprehensively studied and best understood skein module of all.
Let M be an oriented 3-manifold, L f r the set of unoriented framed links (including the empty link ∅) in M up to ambient isotopy, R a commutative ring with unity, and A a xed invertible element in R. Consider the submodule S sub 2,∞ of the free R-module RL f r generated by the Kau man bracket skein relation, L + − AL 0 − A −1 L ∞ , and the trivial component relation, L + (A 2 + A −2 )L, where denotes the trivial framed knot and the skein triple (L + , L 0 , L ∞ ) denotes three framed links in M which are identical except in a small 3-ball in M where they di er as shown:
Kau man bracket skein module (KBSM) of M is de ned as the quotient S 2,∞ (M; R, A) = RL f r /S sub 2,∞ . For brevity, when R = Z[A ±1 ] we use the notation S 2,∞ (M) in the remainder of the paper. One can also work with the relative case in which the oriented 3-manifold (M, ∂M) has 2k marked points {x 1 , x 2 , . . . , x 2k } on ∂M.
e relative Kau man bracket skein module (RKBSM) of (M, ∂M, {x i } 2k 1 ), denoted by S 2,∞ (M, {x i } 2k 1 ; R, A), is the set of all ambient isotopy classes of relative framed links L in (M, ∂M, {x i } 2k 1 ) keeping ∂M xed, such that L ∩ ∂M = ∂L = {x i } 2k 1 , modulo the Kau man bracket relations (see [Prz2] ).
In the year 2000, the second author published the following fundamental theorem about the connected sum of the Kau man bracket skein module of oriented 3-manifolds over the ring Z[A ±1 ] localised by inverting all the cyclotomic polynomials in A. eorem 1.1. [Prz3] If M and N are compact, oriented 3-manifolds, M # N denotes their connected sum, and A k − 1 is invertible in R for any k > 0, then
In particular, the result holds when R = Q(A). is theorem was an an essential tool used in [GJS] to resolve Wi en's niteness conjecture for the KBSM of 3-manifolds over Q(A), since it shows that if S 2,∞ (M; Q(A)) and S 2,∞ (N ; Q(A)) are nite dimensional, then so is S 2,∞ (M # N ; Q(A)), that is, niteness of KBSMs is stable under connected sums. However, eorem 1.1 does not hold when the ring R = Z[A ±1 ]. In this case S 2,∞ (M) is not always nitely generated and o en contains torsion (see [HP2] ). In fact, much less is known about the structure of the Kau man bracket skein module of an oriented 3-manifold when R = Z[A ±1 ] than when R = Q(A). Julien Marché had proposed a conjecture (see [DW] ) about the structure of the KBSM over R = Z[A ±1 ] which was recently disproved by the rst author in [Bak] .
With the goal of understanding the structure of Kau man bracket skein modules over the ring Z[A ±1 ] and giving a complete and detailed description of the KBSM of connected sums and disc sums, the second author had stated the following theorem without proof ( eorem 7.1 in [Prz3] ) about the Kau man bracket skein module over Z[A ±1 ] of the connected sum of handlebodies. eorem 1.2. [Prz3] Let H n denote a genus n handlebody and F 0,n+1 be a disc with n holes so that H n = F 0,n+1 × I . en, S 2,∞ (H n # H m ) = S 2,∞ (H n+m )/I, where I is the ideal generated by expressions z k −A 6 u(z k ), for any even k ≥ 2, and z k ∈ B k (F 0,n+m+1 ), where B k (F 0,n+m+1 ) is composed of links without contractible components and with geometric intersection number k with a disc D separating H n and H m . u(z k ) is a modi cation of z k in the neighbourhood of D, as shown in Figure 1 . e relation z k = A 6 u(z k ), is a result of the sliding relation z k = sl ∂D (z k ) as illustrated in Figure 2 .
In this paper we disprove this theorem by providing a counterexample which is given by H n # H m , n ≥ 2, m ≥ 1 and we show that the ideal I should be replaced by a strictly bigger ideal to obtain the equality in eorem 1.2. At the time of writing this paper the case of H 
In this section we discuss several important properties of Kau man bracket skein modules that are pertinent to this paper, including a description of the KBSM of any 3-manifold using generators and relations. eorem 2.1. [Prz1, Prz2] (1) Let i : M → N be an orientation preserving embedding of 3-manifolds. is yields a homomorphism of skein modules i * :
is correspondence leads to a functor from the category of 3-manifolds and orientation preserving embeddings (up to ambient isotopy) to the category of R-modules with a speci ed invertible element A ∈ R.
(2) If N is obtained from M by adding a 3-handle to M and i : M → N is the associated embedding, then i * :
(3) If M 1 M 2 is the disjoint sum of oriented 3-manifolds M 1 and M 2 then S 2,∞ (M 1 M 2 ; R, A) = S 2,∞ (M 1 ; R, A) ⊗ S 2,∞ (M 2 ; R, A).
(4) ( e Universal Coe cient Property) Let R and R be commutative rings with unity and r : R −→ R be a homomorphism. en the identity map on L f r induces the following isomorphism of R (and R) modules:
e following lemma allows one to write a presentation of the Kau man bracket skein module of any compact oriented 3-manifold using its Heegaard decomposition and knowledge of the presentation of the KBSM of any handlebody. eorem 2.3 describes the KBSM and RKBSM of trivial surface I -bundles and in particular, handlebodies.
Lemma 2.2 (Handle Sliding Lemma). [Prz2, Prz3, HP1] (1) Let M be a 3-manifold with boundary ∂M and γ be a simple closed curve on ∂M. Let N = M γ be the 3-manifold obtained from M by adding a 2-handle along γ and i : M → N be the associated embedding. en i * : S 2,∞ (M; R, A) −→ S 2,∞ (N ; R, A) is an epimorphism. Furthermore, the kernel of i * is generated by the relations yielded by 2-handle sliding. More precisely, if L Here L ∈ L f r en and sl γ (L) is obtained from L by sliding it along γ (that is, we perform 2-handle sliding).
(2) Let M be an oriented compact 3-manifold and consider its Heegaard decomposition (that is, M is obtained from the handlebody H m by adding 2 and 3-handles to it). en the generators of S 2,∞ (M; R, A) are generators of S 2,∞ (H m ; R, A) and the relators of S 2,∞ (M; R, A) are yielded by 2-handle slidings. Prz2] (1) S 2,∞ (F × [0, 1]) is a free module generated by the empty link ∅ and links in F which have no trivial components. Here F is an oriented surface and each link in F is equipped with an arbitrary, but speci c framing. is applies in particular to handlebodies, since H n = F 0,n+1 × I , where H n is a handlebody of genus n and F ,b denotes a surface of genus with b boundary components.
(2) If ∂F = ∅ and all
is a free R-module whose basis is composed of relative links in F without trivial components. e following de nition will be useful in our construction of the counterexample to eorem 1.2.
If W is the 3-manifold obtained by gluing M and N along a part of their boundaries using f then we have the following bilinear form:
Consider the oriented 3-manifold M = H n # H m , n, m ≥ 1 and let D be the compressing disc in M which separates H n and H m . In addition, let γ = ∂D. Now H n # H m is homeomorphic to (H n+m ) γ which is the 3-manifold H n+m with a 2-handle added along γ . Let F 0,n+m+1 be a sphere with n + m + 1 boundary components denoted by a 1 , a 2 , . . . , a n+m+1 . en H n+m ∼ = F 0,n+m+1 × I and we project links in (H n+m ) γ onto F 0,n+m+1 and work with their corresponding link diagrams. e projection of γ onto F 0,n+m+1 is represented by a red line segment as illustrated in Figure 3 . As a special case of De nition 2.4, let W = F 0,n+m+1 × I . Consider a rectangle R, which is the regular neighbourhood of the red line segment, and its embedding under ρ into F 0,n+m+1 = F 0,n+m+1 × { 1 2 }. Choose 2k marked points on the boundary of this rectangle, with k points on the le edge and k points on the right edge. Consider the Temperley-Lieb box M = R × I and relative links in M modulo the Kau man bracket relations. is gives rise to the Temperley-Lieb module T L k which is the relative Kau man bracket skein module of R × I . 1 For any relative multicurve κ in F 0,n+m+1 \ R we have a module homomorphism ρ * : S 2,∞ (R × I , {x i } 2k 1 ) −→ S 2,∞ (H n+m ) as follows: for a given Temperley-Lieb element its image under ρ * is obtained by gluing it to κ along the 2k marked points. See Figure 4 for an example. Let z k be a multicurve in F 0,n+m+1 × { 1 2 } obtained by gluing the identity element Id k of T L k to some κ using the homomorphism ρ * such that z k is in general position with the compressing disc D having geometric crossing number k with it (see Figure 4a ). In (H n+m ) γ consider the 2-handle slidings of z k along γ described in Figures 5 and 6 . ese handle slidings have support in (R × I ) γ , the Temperley-Lieb box with a 2-handle a ached along γ . in terms of the Temperley-Lieb elements. By extension, using De nition 2.4 and embedding T L k into H n+m as described earlier, we get the relations in S 2,∞ (H n+m ) γ .
In this section we construct a counterexample to eorem 1.2. Our result is summarised as follows: eorem 4.1.
(1) S 2,∞ (H 2 # H 1 ) = S 2,∞ (H 3 )/I. (2) In general, S 2,∞ (H n # H m ) = S 2,∞ (H n+m )/I, where n + m ≥ 3.
Proof. We rst prove part (1) of the theorem. Part (2) follows by an easy generalisation. Consider the oriented 3-manifold H 2 # H 1 and the positive 2-handle sliding ϕ on the bo om arc illustrated F 8. Curve system in F 0,4 leading to the counterexample in Figure 6 . We compute w(Id k ) recursively starting from k = 2, in which case we obtain the following result.
(1) w(Id 2 ) = A 2 Id 2 + (1 − A −4 )e 1 = A 2 Id 2 + A −4 (A 4 − 1)e 1 .
In general, from Figure 7 , we get the following recursive relation for w(Id k ):
Here w(Id k ) denotes the mirror image of w(Id k ) (see Figure 6b ). In particular, when k = 3 we get the following equation:
(3) w(Id 3 ) = A 4 Id 3 + (A 2 − A −6 )e 2 + (A 2 − A −2 )e 1 + (1 − A −4 )(e 1 e 2 + e 2 e 1 ), and when k = 4, we get the following equation using Equations (2) and (3) :
(4) w(Id 4 ) = A 6 Id 4 + (A 4 − 1)e 1 + (A 4 − A −4 )e 2 + (A 2 − A −2 )(e 1 e 2 + e 2 e 1 ) + (A 4 − A −8 )e 3 + (A 2 − A −6 )(e 1 e 3 + e 2 e 3 + e 3 e 2 ) + (1 − A −4 )(e 1 e 2 e 3 + e 3 e 2 e 1 + e 1 e 3 e 2 + e 2 e 3 e 1 ).
Since ϕ (Id 4 ) = A 6 w(Id 4 ), therefore, (5) ϕ l (Id 4 ) = A 12 Id 4 + (A 10 − A 6 )e 1 + (A 10 − A −2 )e 3 + +(A 10 − A 2 )e 2 + (A 8 − A 4 )(e 2 e 1 + e 1 e 2 ) +(A 8 − 1)[e 2 e 3 + e 3 e 2 + e 1 e 3 ] + (A 6 − A 2 )(e 3 e 2 e 1 + e 1 e 3 e 2 + e 2 e 3 e 1 + e 1 e 2 e 3 ).
Consider the handle sliding relation ϕ l (Id 4 ) ≡ Id 4 in the relative Kau man bracket skein module of (R × I ) γ . erefore, (6) (1 − A 12 )Id 4 ≡ (A 10 − A 6 )e 1 + (A 10 − A −2 )e 3 + +(A 10 − A 2 )e 2 + (A 8 − A 4 )(e 2 e 1 + e 1 e 2 ) + (A 8 − 1)[e 2 e 3 + e 3 e 2 + e 1 e 3 ] + (A 6 − A 2 )(e 3 e 2 e 1 + e 1 e 3 e 2 + e 2 e 3 e 1 + e 1 e 2 e 3 ).
By performing 2-handle sliding ϕ t on the upper string, the roles of e 1 and e 3 are exchanged in the above relation and we get the following:
(7) (1 − A 12 )Id 4 ≡ (A 10 − A −2 )e 1 + (A 10 − A 6 )e 3 + (A 10 − A 2 )e 2 + (A 8 − A 4 )(e 2 e 3 + e 3 e 2 ) + (A 8 − 1)(e 2 e 1 + e 1 e 2 + e 1 e 3 ) + (A 6 − A 2 )(e 1 e 2 e 3 + e 1 e 3 e 2 + e 2 e 3 e 1 + e 3 e 2 e 1 ).
Subtracting the Equation (6) from Equation (7) we get: 0 ≡ (A 6 − A −2 )(e 1 − e 3 ) + (A 4 − 1)(e 2 e 1 + e 1 e 2 − e 2 e 3 − e 3 e 2 ), and thus,
0 ≡ A −2 (A 8 − 1)(e 1 − e 3 ) + (A 4 − 1)(e 2 e 1 + e 1 e 2 − e 2 e 3 − e 3 e 2 ).
Every Temperley-Lieb element in the equation above intersects the compressing disc D transversely twice. erefore, we can use Equation (1) in this situation by carefully taking into account which strings intersect D. For example, in the rst relation below, the third and fourth strings intersect with D. us, we get the following equivalences: 
We use the equivalence (A 8 − 1)e 1 ≡ (A 2 −A 6 )e 1 e 3 ≡ (A 8 − 1)e 3 and therefore, the rst two terms in Equation (8) cancel out and we get the following relation:
(10) 0 ≡ (A 4 − 1)(e 2 e 1 − e 2 e 3 + e 1 e 2 − e 3 e 2 ).
We now embed T L 4 into H 3 = F 0,4 × I as described earlier and under the homomorphism ρ * , we get that e 1 e 2 → a 1 a 3 [a 2 a 3 ], e 3 e 2 → a 2 a 3 [a 1 a 3 ], e 2 e 1 → [a 1 a 2 ], and e 2 e 3 → [a 1 a 2 ]. In particular, ρ * (e 2 e 1 ) = ρ * (e 2 e 3 ) as illustrated in Figure 9 . Here [a i a j ] represents a curve that separates the boundary components a i and a j from the other two boundary components of F 0,4 . [a 1 a 2 ] F 9. . us, in S 2,∞ (H 3 ) γ , Equation (10) results in the following equivalence:
(11) 0 ≡ (A 4 − 1)(a 1 a 3 [a 2 a 3 ] − a 2 a 3 [a 1 a 3 ]).
is relation consists of two curve systems that are not ambient isotopic in F 0,4 . Notice that in eorem 1.2 the ideal I has the following generators: for every even k and z k having minimal intersection number k with D, the ideal I has exactly one generator (A 2k+4 − 1)z k + k−2 i=0 α i (A)z i . erefore, the right hand side of Equation (11) is not contained in the ideal I and thus, we have found a new relation in S 2,∞ (H 3 ) γ which serves as a counterexample to eorem 1.2. is completes the proof of part (1) of eorem 4.1.
To prove part (2) of eorem 4.1, we observe that H 2 # H 1 can be embedded in the connected sum of any two handlebodies of higher genera and the same curve system in Figure 9 embedded in the surface F 0,n+m+1 leads to a counterexample for all connected sums H n # H m , n + m ≥ 3.
Remark 4.2. When we compare the sliding relations ϕ t with ϕ t we obtain the equivalence 0 ≡ (A 4 − 1) 2 (e 1 + e 3 − e 1 e 2 e 3 − e 3 e 2 e 1 ) (see the calculation below). For H n # H 1 this relation cannot give a counterexample as it vanishes a er embedding the Temperley-Lieb box into H n # H 1 since ρ * (e 1 ) = ρ * (e 3 ) = ρ * (e 1 e 2 e 3 ) = ρ * (e 3 e 2 e 1 ) (see Figure 10 ). However, this relation is still nontrivial if we embed the Temperley-Lieb box into H n # H 2 as in Figure 4a . We leave this as an exercise to the reader. Calculation: Consider the sliding relation given by ϕ t in Equation (7). Multiplying the sliding relation given by its mirror image ϕ t by A 12 and adding it to Equation (7) we get:
(12) 0 ≡ A 2 (A 4 − 1) 2 (e 3 − e 3 e 2 e 1 − e 1 e 2 e 3 − e 3 e 1 e 2 − e 2 e 1 e 3 ) + A −2 (A 12 − 1)(1 − A 4 )e 1 + (A 8 − 1)(1 − A 4 )(e 1 e 3 + e 2 e 1 + e 1 e 2 ). Now the terms which intersect the compressing disc in exactly two points (for example, e 1 , e 3 , e 1 e 2 , e 2 e 1 , e 1 e 2 e 3 , and e 3 e 2 e 1 ) satisfy Equation (9). ree terms e 1 e 3 , e 1 e 3 e 2 , and e 2 e 3 e 1 are disjoint from compressing disc. us, a er reduction we get the required equivalence:
We have shown that eorem 1.2 does not hold in full generality. However, our calculations suggest that the sliding relations that generate the ideal I are enough in the case of H 1 # H 1 .
Conjecture 5.1. S 2,∞ (H 1 # H 1 ) = S 2,∞ (H 2 )/I.
In support of the conjecture we have checked that when k = 4, all the handle sliding relations come from ϕ t and sliding relations from the case k = 2, and when k = 6, all the handle sliding relations again come from ϕ t and sliding relations from the smaller cases k = 2 and 4.
In a future paper we plan to resolve this conjecture and as an application use it to compute the Kau man bracket skein module of the connected sum of lens spaces over the ring Z[A ±1 ]. In particular, we will compare our result with the result in [Mro] about the connected sum of two copies of the real projective space, RP 3 # RP 3 . 6. A e second author was partially supported by Simons Collaboration Grant-637794 and the CCAS Enhanced Travel award. e authors would like to thank Charles Frohman due to whom they decided to provide a proof for eorem 1.1 and ended up disproving it. 2