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A HOMFLY‐PT type invariant for integral homology 3‐spheres

Shunsuke Tsuji
Research Institute for Mathematical Sciences, Kyoto University
In this paper, we will introduce how to construct some invariants for integral homology
3‐spheres, using skein algebras.

1

Skein algebras

Let \Sigma be a compact connected oriented surface with nonempty boundary. We fix basepoint sets J^{-}, J^{+} and J of \partial\Sigma such that J^{-}\cap J^{+}=\emptyset and \# J^{-}=\# J^{+} . We denote by
S(\Sigma, J) the Kauffman bracket skein module, which is the quotient of the set of the free\mathbb{Q}[[A+1]] ‐module with basis the set of the tangles in \Sigma\cross I with basis J by the relation
shown in Fig. 2. Here, we denote by I the closed interval between 0 and 1. We denote by\mathcal{A}(\Sigma, J^{-}, J^{+}) the HOMFLY‐PT skein module which is the quotient of the set of the free\mathbb{Q}[\rho][[h]] ‐module with basis the set of the tangles in \Sigma\cross I from J^{-} to J^{+} by the relation
shown in Fig. 1. See, for example, [7] p.21 Definition 3.2 and [10] p.9 Definition 3.2.
We simply denote S(2)def=S(\Sigma, \emptyset) and \mathcal{A}(\Sigma)def=\mathcal{A}(\Sigma, \emptyset) . Let e_{1} and e_{2} be embeddings
from \Sigma\cross I into \Sigma\cross I defined by e_{1}(x, t)=(x, \frac{t+1}{2}) and e_{2}(x, t)=(x, \frac{t}{2}) . For x, x'\in \mathfrak{g}and z\in \mathfrak{g}_{M} , xx’, xy and yx is defined by e_{1}(x)\cup e_{2}(x'), e_{1}(x)\cup e_{2}(y) and e_{1}(y)\cup e_{2}(x)where (\mathfrak{g}, \mathfrak{g}_{M})=(\mathcal{S}(\Sigma), S(\Sigma, J)), (\mathcal{A}(\Sigma), \mathcal{A}(\Sigma, J^{-}, J^{+})) . We defined a Lie bracket [x, x']
and a Lie action \sigma(x)(z) satisfying \epsilon_{\mathfrak{g}}[x, x']=xx' -x'x and \epsilon_{\mathfrak{g}}\sigma(x)(z)=xz-xz where\epsilon_{S(\Sigma)}=-A+A^{-1} and \epsilon_{A(\Sigma)}=h . For details, see [7] p.23 Definition 3.8. and [10]p.16 Definition 3.13. We can consider some skein algebras and there exists some Liehomomorphisms around skein algebras.

*\ovalbox{\t \smal REJECT}_{\backslash }\backslash -\nwar ow_{/} ^{\prime\paral el}/=h\lambda(\zeta\backslash /_{\backslash }^{\backslash }\backslash =A) (・ |\sqrt{}‐ı \bigwedge_{\wedge}\smile( C_{-}^{>})) =\frac{2\sinh(\rho h)}{h}\emptyset O=(-A^{2}-A^{-2})\emptyset\mathscr{M}^{\backslah\backslah}=\exp(\rho h)Fig. 1: \mathcal{A}(\Sigma)+A

Fig. 2: \mathcal{S}(\Sigma)

We fix two base points *1, *2\in\partial\Sigma . We simply denote \mathcal{A}(\Sigma, *1, *2)=\mathcal{A}(\Sigma def, \{*1\}, \{*2\})and S(\Sigma, *1, *2)=S(\Sigma def, \{*1, *2\}) .

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1.1

\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} :\mathcal{A}(\Sigma)ar ow S(\mathbb{Q}\hat {\pi}_{1}(\Sigma))

Let \hat{\pi}_{1}(\Sigma) be the set of the conjugacy classes of \pi_{1}(\Sigma) . The quotient map is denoted by |\cdot| :\mathbb{Q}\pi_{1}(\Sigma)ar ow \mathbb{Q}\hat{\pi}_{1}(\Sigma) . Goldman defined the Lie bracket [\cdot, \cdot] : \mathbb{Q}\hat{\pi}_{1}(\Sigma)\cross \mathbb{Q}\hat{\pi}_{1}(\Sigma)ar ow \mathbb{Q}\hat{\pi}_{1}(\Sigma) in[1]. Kawazumi‐Kuno defined the Lie action \sigma : \mathbb{Q}\hat{\pi}_{1}(\Sigma)\cross \mathbb{Q}\pi_{1}(\Sigma, *1, *2)arrow \mathbb{Q}\pi_{1}(\Sigma, *1, *2)
in [3]. Let S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))def=\oplus_{i=0}S^{i}(\mathbb{Q}\hat{\pi}_{1} (\Sigma)) be the symmetric tensor algebra of \mathbb{Q}\hat{\pi}_{1}(\Sigma)
over \mathb {Q} . The Leibniz rule defines the Lie bracket [\cdot, \cdot] : S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\cross S(\mathbb{Q}\hat{\pi}_{1}(\Sigma)) arrow
S(\mathbb{Q}\hat{\pi}_{1}(\Sigma)) . The Lie action \sigma(\cdot)(\cdot) : S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\cross S(\mathbb{Q}\hat{\pi}_{1}(\Sigma)) \otimes \mathbb{Q}\pi_{1}(\Sigma, *1, *2)arrow S(\mathbb{Q}\hat{\pi}_{1}(\Sigma) )\otimes\mathbb{Q}\pi_{1}(\Sigma, *1, *2) is defined by \sigma(v_{1}\cdots v_{j})(V_{1}\otimes\gamma)^{def}=[v_{1}\cdots v_{j}, V_{1}] \otimes\gamma+V_{1}\cdot(\sum_{j=1}^{j}v_{1}\cdots v_{j'-1}.

v_{j'+1}\cdot v_{j}\otimes a(v_{j})(7)) . Let L be an oriented framed link in \Sigma\cross I having connected componentsl_{1} ,
, l_{j} . Then, we put \psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (L)def=[l_{1}][l_{j}]\in S^{j}(\mathbb{Q}\hat{\pi}_{1}(\Sigma)) . Let T be an oriented

framed tangle from *1to*2 having closed components l_{1}, l_{2} ,, l_{j} and non‐closed compo‐

nents r . Then, we also put \psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (T)def=[l_{1}]\cdots[l_{j}]\otimes[r]\in S^{j}(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\otimes \mathbb{Q}\pi_{1}(\Sigma, *1, *2) .
The maps \psi_{Aar ow \mathb {Q}\hat{\pi}_{ \imath} induces the \mathb {Q} ‐linear maps

\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} :\mathcal{A}(\Sigma)ar ow S(\mathbb{Q}\hat {\pi}_{1}(\Sigma))\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1}}:\mathcal{A}(\Sigma, * )arrow S(\mathbb{Q} \hat{\pi}_{1}(\Sigma))\otimes \mathbb{Q}\pi_{1}(\Sigma, * )satisfying[(\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (x), \psi_{Aarrow \mathbb{Q}\hat{\pi} _{1} )(x')]=\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} ([x, x']),\sigma(\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (x) (\psi_{Aarrow \mathbb{Q} \hat{\pi}_{1} (y) =\psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (\sigma(x)(y)for any x, x'\in \mathcal{A}(\Sigma) and y\in \mathcal{A}(\Sigma, *1, *2) .
1.2\mathcal{A}(\Sigma)=\oplus_{x\in H_{1}(\Sigma,Z)}\mathcal{A}_{x}(\Sigma)

Let L_{+}, L_{-} and L_{0} be three framed links which are differ only in a closed ball as shownin the first, second and third terms of the first equation. The homology classes of L_{+},
L_{-} and L_{0} equal each other. Let L and L' be two framed links which are differ only ina closed ball as shown in the first and second terms of the second equation in Fig. 1.
Then, the homology classes of L equals the one of L' . Hence, there exists a direct sum
decomposition

Then we have\mathcal{A}(\Sigma)=\bigoplus_{x\inH_{1}(\Sigma,\mathb {Z}) \mathcal{A}_{x} (\Sigma).

\mathcal{A}_{x}(\Sigma)\cdot \mathcal{A}_{y}(\Sigma)\subset \mathcal{A}_{x+y} (\Sigma)[\mathcal{A}_{x}(\Sigma), \mathcal{A}_{y}(\Sigma)]\subset \mathcal{A}_{x+y} (\Sigma)
for any x, y\in H_{1}(\Sigma, \mathbb{Z}) .
1.3\psi_{Aarrow S}:\mathcal{A}_{0}(\Sigma)arrow \mathcal{S}(\Sigma)

We put \psi_{Aarrow S}' by the Q ‐module homomorphism from \mathcal{A}(\Sigma, J^{-}, J^{+}) to S(\Sigma, J^{-}\cup J^{+})
defined by T\mapsto(-A)^{w(T)}T, h\mapsto-A^{2}+A^{-2} and \exp(\rho h)=A^{4} . Here w(T)def=

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Fig. 3: T_{1}T_{2}, |T|, r_{1}r_{2}#{positive crossings of T } -\# {negative crossings of T } is the writhe number of T . By [10]Proposition 7.15, \psi_{Aarrow S}' is well‐defined. We remark that \psi_{Aarrow S}' is not a Lie algebra ho‐
momorphism. The Q‐module homomorphism \psi_{Aarrow S} is defined by \psi_{Aarrow s=}^{def}\frac{1}{A+A^{-1} \psi_{Aarrow S}'.
Then, \psi_{Aarrow S}:\mathcal{A}_{0}(\Sigma)arrow \mathcal{S}(\Sigma) is a Lie algebra homomorphism.
2

Formulas for Dehn twists

We denote by \pi_{1}^{+}(\Sigma, *1, *2) the set of the regular homotopy classes of free immersed paths
from *1 to *2 . We fix a simple path r_{*}\in\pi_{1}^{+}(\Sigma, *1, *2) . A composite . : \pi_{1}^{+}(\Sigma, *1, *2)\cross\pi_{1}^{+}(\Sigma, *1, *2)arrow\pi_{1}^{+}(\Sigma, *1, *2) is defined by Fig. 3. We have r\cdot r_{*}=r_{*}\cdot r=r for any r\in\pi_{1}^{+}(\Sigma, *1, *2) . For an immersed path \varsigma : Iarrow\Sigma aframed oriented tangle \varphi(\varsigma) is defined\epsilon>0\varphi(\varsigma) I\cross Iarrow\Sigma\cross I,
This \varphi induces a \mathbb{Q}[\rho][[h]] ‐module homomorphism \varphi : \mathcal{P}(\Sigma, *1, *2)arrow \mathcal{A}(\Sigma, *1, *2) where\mathcal{P}(\Sigma, *1, *2) is the quotient of the free \mathbb{Q}[\rho][[h]] ‐module with basis \pi_{1}^{+}(\Sigma, *1, *2) by the
relation which is the second equation of Fig. 1. A composit \mathcal{A}(\Sigma, *1, *2)\cross \mathcal{A}(\Sigma, *1, *2)arrowby:(t, s) \mapsto(\varsigma(t), \frac{2-t+\epsilon s}{3})’whereis anumber small enough.

\mathcal{A}(\Sigma, *1, *2) is defined by Fig. 3. Then, the \mathbb{Q}[\rho][[h]] ]‐module homomorphism is a \mathbb{Q}[\rho][[h]]-algebra homomorphism. We define the closure \mathcal{A}(\Sigma, *1, *2)arrow \mathcal{A}(\Sigma) by Fig. 3.
2.1

Filtrations

In this subsection, we will introduce some filtrations of skein algebras and skein mod‐ules. An augmentation map aug : \mathcal{P}(\Sigma, *1, *2)arrow \mathbb{Q}[\rho] is defined by aug(r)=1 andaug(h)=0 for r\in\pi_{1}^{+}(\Sigma, *1, *2) . Let F^{n}\mathcal{P}(\Sigma, *1, *2) be a \mathb {Q} ‐linear subspace gener‐
ated by \sum_{2i+\dot{J}\geq n}h^{i}(keraug)^{j} . For any n\in \mathbb{Z}_{\geq 0} , the filtrations \{F^{n}\mathcal{A}(\Sigma, *1, *2)\}_{n\geq 0} and\{F^{n}\mathcal{A}(\Sigma)\}_{n\geq 0} are defined by

F^{n} \mathcal{A}(\Sigma, *, *)^{def}=.\sum_{i_{1}+\dot{i}_{2}+\cdot\geq n}. \varphi 1,2(F^{i_{2} \mathcal{P}(\Sigma, * ) |F^{n} \mathcal{A}(\Sigma)^{def}=.\sum_{i_{1}+\cdots\geq n}(F^{i_{1} \mathcal{P}(\Sigma, * (F^{i_{2} \mathcal{P}(\Sigma, * ) |\cdots

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These filtrations \{F^{n}\mathcal{A}(\Sigma, *1, *2)\}_{n\geq 0} and \{F^{n}\mathcal{A}(\Sigma)\}_{n\geq 0} induces the filtrations \{F^{n}S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\}_{n\geq 0},\{F^{n}S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\otimes \mathbb{Q}\pi_{11,2}(\Sigma, * *)\}_{n\geq 0}, \{F^{n}\mathbb{Q}\hat{\pi}_{1}(\Sigma)\}_{n\geq 0}, \{F^{n}\mathbb{Q}\pi_{11,2}(\Sigma, * )\}_{n\geq 0}, \{F^{n}S(\Sigma)\}_{n\geq 0}

and \{F^{n}S(\Sigma, *1, *2)\}_{n\geq 0} byF^{n}S( \sum)def=. \psi_{Aarrow S}(F^{n}\mathcal{A}(\sum)), F^{n}S( \sum, *1, *2)def=. \psi_{Aarrow S}(F^{n}\mathcal{A}(\sum, *1, *2))F^{n}S( \mathbb{Q}\hat{\pi}_{1}(\sum))def=. \psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (F^{n}\mathcal{A}(\sum)), F^{n}((S( \mathbb{Q}\hat{\pi}_{1}(\sum))\otimes \mathbb{Q}\pi_{1}(\sum, *1, *2) ))def=. \psi_{Aarrow \mathbb{Q}\hat{\pi}_{1} (F^{n}\mathcal{A}(\sum, *1, *2))F^{n} \mathbb{Q}\hat{\pi}_{1}(\sum)def=. \mathbb{Q}\hat{\pi}_{1}(\sum)\cap S(\mathbb{Q}\hat{\pi}_{1}(\sum))F^{n} \mathbb{Q}\pi_{1}(\sum, *1, *2)def=. \mathbb{Q}\pi_{1}(\sum, *1, *2)\cap F^{n}(S(\mathbb{Q}\hat{\pi}_{1}(\sum)) \otimes \mathbb{Q}\pi_{1}(\sum, *1, *2))\cdot,,

These filtrations satisfy[F^{i}\mathfrak{g}, F^{j}\mathfrak{g}]\subset F^{i+j-2}\mathfrak{g}, \sigma(F^{i}\mathfrak{g})(F^{j}\mathfrak{g}_{M})\subset F^{i+j-2}\mathfrak{g} _{M}for (\mathfrak{g}, \mathfrak{g}_{M})=(\mathcal{A}(\Sigma), \mathcal{A}(\Sigma, *1, *2)), (S(\Sigma), S(\Sigma, *1, *2)), (S(\mathbb{Q}\hat{\pi}_{1}(\Sigma)), S(\mathbb{Q}\hat{\pi}_{1}(\Sigma))\otimes \mathbb{Q}\pi_{1}(\Sigma, *1, *2)) ,(\mathbb{Q}\hat{\pi}_{1}(\Sigma), \mathbb{Q}\pi_{1}(\Sigma, *1, *2)) . For \mathfrak{g}=\mathcal{P}, \mathcal{A}, S, \mathbb{Q}\hat{\pi}_{1}, \mathbb{Q}\pi_{1} , Their completion is denoted by\hat{\mathfrak{g}}1,2defFiarrow\infty 1,21,2,\hat{\mathfrak{g} (\Sigma)^{def}=. \Psi ar ow\infty \mathfrak{g}(\Sigma)/F^{i} \mathfrak{g}(\Sigma).

2.2

A formula for Dehn twists on \mathbb{Q}\hat{\pi}_{1}(\Sigma)

Kawazumi‐Kuno and Massuyeau‐Turaev give a formula for the action of Dehn twists on\widehat{\mathbb{Q}\pi_{1} (\Sigma, *1, *2) . For a simple closed curve , they put L_{\mathbb{Q}\hat{\pi}_{1} (c) def=\frac{1}{2}|(\log(r_{c}) ^{2}| where r_{c} iscan element of \pi_{1}(\Sigma) such that|r_{c}|=c.

Theorem 2.1 ([2] [3] [5]). For a Dehn twist t_{c} along c , we obtaint_{c}(\cdot)=\exp(\sigma(L_{\mathbb{Q}^{-} \pi(c)))(\cdot)^{def}=.
2.3\sum_{i=0}^{\infty}\frac{1}{\dot{i}! (\sigma(L_{\mathb {Q}\hat{\pi} (c) ^{i}( \cdot):\widehat{\mathb {Q}\pi_{1 ,2} (\Sigma,*)ar ow\widehat{\mathb {Q}\pi_{1} }(\Sigma,*).

A formula for Dehn twists on \mathcal{A}(\Sigma)

We denote by \varphi_{n} the Q ‐module homomorphism from (\hat{\mathcal{P} (\Sigma, *1, *2) ^{\otimes n}\wedge to \hat{\mathcal{A} (\Sigma, *1, *2) de‐
fined by \varphi_{n} (r_{1}\otimes \otimes r_{n})=|\varphi(r_{1})|\cdots|\varphi(r_{n})| . By induction, we define \lambda^{[n]}(X_{1} , X_{n})by \lambda^{[1]}(X_{1})=\frac{1}{2X_{1}}(\log(X_{1}))^{2} and\lambda^{[n+1]} (X_{1}, \cdot \cdot \cdot X_{n+1})^{def}=. \frac{\lambda^{[n]} (X_{1},X_{n})-\lambda^{[n]}(X_{2},\cdots,X_{n+1})}{X_{1}-X_{n+1} .
For a simple closed curve c , let r_{c} be an element of \pi_{1}^{+}(\Sigma, *1, *2) such that |\varphi(r_{c})| is anelement of \mathcal{A}(\Sigma) represented by a knot presented by a diagram c . We put

L_{A}(c)^{def}=.

( \frac{h/2}{arcsinh(h/2)})^{2}(\sum_{n=1}^{\infty}\frac{h^{n-1}\exp(n\rho h)} {n}\varphi_{n}(r_{1,n}\cdots r_{n,n}\lambda^{[n]}(r_{1,n}, \cdots, r_{n,n}) - \frac{1}{3}p^{3}h^{2})

,

,

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where r_{i,n}=r_{*}\otimes\exp(-ph)r_{c}\otimes r_{*}def\otimes(i-1)\otimes n-i . Here, for F(X_{1}, X_{2}, \cdots , X_{n})=\sum a_{i_{1},i_{2},\cdots,i_{n}}(X_{1}, x_{n}\in1)^{i_{1}}(X_{2}-1)^{i_{2}}\cdots(X_{n}-1)^{i_{n}}\in \mathbb{Q}[[X_{1}-1, X_{2}-1, , X_{n}-1]] and x_{1}, x_{2} ,r_{*}^{\otimes n}+(keraug)^{\otimes n} , we define

F (x_{1}, x_{2} , x_{n})= \sum a_{i_{1},i_{2},\cdots,i_{n} (x_{1}-r_{*} ^{\otimes n})^{i_{1} (x_{2}-r_{*}^{\otimes n})^{i_{2} \cdots(x_{n}-r_{*} ^{\otimes n})^{i_{n} .
Here r^{Xn} is a unit of\hat{\mathcal{P} (\Sigma, *1, *2)^{\otimes n}\wedge.

Theorem 2.2 ([10] p.29 Theorem 5.2). The element L_{A}(c) is well‐defined.
Theorem 2.3 ( [10] p.28 Theorem 5.1). For a Dehn twist t_{c} along c , we obtaint_{c}(\cdot)=\exp(\sigma(L_{A}(c)))(\cdot)^{def}=.
2.4\sum_{\dot{i}=0}^{\infty}\frac{1}{i!}(\sigma(L(c) ^{i}(\cdot): \hat{\mathcal{A} (\Sigma, * )ar ow\hat{\mathcal{A} (\Sigma, * ).

A formula for Dehn twists on \mathcal{S}(\Sigma)

For a simple closed curve c , we denote by

L_{S}(c)^{def}= \frac{-A+A^{-1} {4\log(-A)}( arccosh( \frac{-c}{2} ) )^{2}-(-A+A^{-1})\log(-A) .
Here we also denote by c an element of S(\Sigma) represented by a knot presented by a diagramc.

Theorem 2.4 ( [7] p.26 Theorem 4.1). For a Dehn twist t_{c} along c , we obtaint_{c}( \cdot)=\exp(\sigma(L_{S}(c) )(\cdot)^{def}=. \sum_{\dot{i}=0}^{\infty} \frac{1}{i!}(\sigma(L_{S1,2}(c) )^{i}(\cdot):\hat{S}(\Sigma, * )ar ow\hat{S} (\Sigma, * )3.

Embeddings of the Torelli group into ( F^{3}\hat{\mathfrak{g} (\Sigma_{g,1}) , bch) for \mathfrak{g}=\mathbb{Q}\hat{\pi}_{1}, S, \mathcal{A}We call the kernel of the action the mapping class group of \Sigma_{g,1} on the homology classof \Sigma_{g,1} the Torelli group. The Torelli group \mathcal{I}(\Sigma_{g,1}) is generated by t_{c_{1} t_{c_{2}^{-1} for a pair ofdisjoint simple closed curves (c_{1}, c_{2}) bounding a surface, which is called by a bounding pair.
We can consider F^{3}\hat{\mathfrak{g} (\Sigma_{g,1}) as the group where the composition is the Baker‐Campbell‐
Hausdorff series, which is simply denote by bch(\cdot, \cdot) . We can consider F^{3}\hat{\mathfrak{g} (\Sigma_{g,1}) as thegroup whose composition is the Baker‐Campbell‐Hausdorff series, which is simply denoteby bch(\cdot, \cdot) .

Theorem 3.1 (\mathfrak{g}=\mathbb{Q}\hat{\pi}_{1} : [2] [3], \mathfrak{g}=S : [8] p.15 Theorem 3.13, p.16 Corollary 3.15, \mathfrak{g}=\mathcal{A} :[10] p.54 Theorem 7.13, Corollary 7.14. ). The group homomorphism \zeta_{\mathfrak{g} from \mathcal{I}(\Sigma_{g,1}) to( F^{3}\mathfrak{g}(\Sigma_{g,1}) , bch) defined by \zeta(t_{c_{1}}t_{c_{2}^{-1}})=L_{\mathfrak{g}}(c_{1})-L_{\mathfrak{g}}(c_{2}) for any bounding pair (c_{1}, c_{2}) is
well‐defined and injective. Furthermore, for \xi\in \mathcal{I}(\Sigma_{g,1}) , we have \xi(\cdot)=\exp(\sigma(\zeta_{\mathfrak{g}}(\xi)))(\cdot) :\hat{\mathfrak{g}}(\Sigma, *, *)arrow\hat{\mathfrak{g}}(\Sigma, *, *).

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Let \{\mathcal{I}^{(n)}(\Sigma_{g,1})\}_{n\geq 1} be the lower central series, defined by \mathcal{I}^{(1)}(\Sigma_{g,1})def=\mathcal{I}(\Sigma_{g,1}) and\mathcal{I}^{(n+1)}(\Sigma_{g,1})def= [\mathcal{I}(\Sigma_{g,1}),\mathcal{I}^{(n)}(\Sigma_{g,1})] .

Using the embedding \zeta_{\mathfrak{g} :\mathcal{I}(\Sigma_{g,1})arrow F^{3}\mathfrak{g}(\Sigma_{g,1}) ,we can define a filtration \{\mathcal{M}_{\mathfrak{g} ^{(n)}\}_{n\geq 1} of \mathcal{I}(\Sigma_{g,1}) by \mathcal{M}_{\mathfrak{g} ^{(n)}(\Sigma_{g,1})def=\zeta_{\mathfrak{g} ^{-1} (F^{n+2}\hat{\mathfrak{g} (\Sigma_{g,1}) for\mathfrak{g}=\mathbb{Q}\hat{\pi}_{1}, S, \mathcal{A} . We call \{\mathcal{M}_{\mathb {Q}\hat{\pi}_{1} ^{(n)}\}_{n\geq 1} the Johnson filtration. They satisfy\mathcal{I}^{(n)}(\Sigma_{g,1})\subset \mathcal{M}_{A}^{(n)}(\Sigma_{g,1}) \subset \mathcal{M}_{S}^{(n)}(\Sigma_{g,1})\cap \mathcal{M}_{\mathbb{Q}\hat{\pi} _{ \imath} }^{(n)}(\Sigma_{g,1})
4.

Invariants for integral homology 3‐spheres

In this subsection, the symbol \mathfrak{g} is a symbol \mathcal{S} or a symbol \mathcal{A} . Using the embedding\zeta_{\mathfrak{g} : \mathcal{I}ar ow F^{3}\hat{\mathfrak{g} (\Sigma_{g,1}) we obtain an invariant for integral homology 3‐spheres.

Theorem 4.1 ( \mathfrak{g}=S : [9] p.1 Theorem 1.1, \mathfrak{g}=\mathcal{A} : [10] p.60 Theorem 9.1. ). Fix a

Heegaard splitting of S^{3}=H_{g}^{+} \bigcup_{\iota}H_{g}^{-} where H_{g}^{+} and H_{g}^{-} are handle bodies of genus g and\iota is a diffeomorphism from\partial H_{g}^{+} to \partial H_{g}^{-} . We consider \Sigma_{g,1} as the closure of \partial H_{g}^{+} except
for a closed disk. Let e:\Sigma_{g,1}arrow S^{3} be a tubular neighborhood inducing e_{*} : \hat{\mathfrak{g} (\Sigma_{g,1})ar ow R_{\mathfrak{g}where R_{s=}^{def}\mathbb{Q}[[A+1]] and R_{A^{def}}=\mathbb{Q}[\rho][[h]] . Then, the map Z_{\mathfrak{g} : \mathcal{I}(\Sigma_{g,1})ar ow R_{\mathfrak{g} defined by

Z_{\mathfrak{g} ( \xi)^{def}=\sum_{\dot{i}=1}^{\infty}\frac{1}{\dot{i} !\epsilon_{\mathfrak{g} ^{i} e*( \zeta_{\mathfrak{g} (\xi) ^{i}) induces a map z_{\mathfrak{g} : \mathcal{H}^{def}= {integral homology 3 spheres}

R_{\mathfrak{g} , H_{g}^{+} \bigcup_{b\circ\xi}H_{g}^{-}\mapsto Z(\xi)

arrow

.

We denotez^{s1_{2}}(M)=1+z_{1}^{s{\imath}_{2}}(M)(q-1)+z_{2}^{s{\imath}_{2}}(M)(q-1)^{2} +the invariant of an integral homology 3‐sphere
denoteMdefined by T. Ohtsuki [6]. We alsoz^{s{\imath}_{N}}(M)=1+z_{1}^{s{\imath}_{N}}(M)(q-1)+z_{2}^{s{\imath}_{N}}(M)(q -1)^{2}+
using the s1_{N} ‐quantum group in [4].
The theorem will appear in our paper in preparation.

Theorem 4.2 ([11]). We obtain z^{s1_{2} (\cdot)=z_{S}(\cdot)_{|A^{4}=q}=z_{A}(\cdot)_{|\exp(\rho h)=q,h=- q^{\frac{1}{2} +q^{-\frac{1}{2} } andz^{s{\imath}_{N} (\cdot)=z_{A}(\cdot)_{1}7\exp(\rho h)=q^{N/2},h=-q^{1{\imath} \Sigma+q^{-2}.
Since e_{*}(F^{2n-1}\mathfrak{g})\subset(\epsilon_{\mathfrak{g} ^{n}) , we havez_{\mathfrak{g} (H_{g}^{+} \bigcup_{\iota 0\xi}H_{g}^{-})\in 1+ (\epsilon_{\mathfrak{g} ^{n})for \xi\in \mathcal{M}_{\mathfrak{g} ^{(2n-1)} . Hence, if \xi\in \mathcal{M}_{\mathcal{S} ^{(2n-1)} , we havez^{s1_{2}}(H_{g}^{+} \bigcup_{eo\xi}H_{g}^{-})\in 1+((q-1)^{n}).

Furthermore, if \xi\in \mathcal{M}_{A}^{(2n-1)} , we havez^{s{\imath}_{N}}(H_{g}^{+} \bigcup_{\iota\circ\xi}H_{g}^{-})\in 1+((q-1)^{n})for anyN.

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[8] S. Tsuji, An action of the Torelli group on the Kauffman bracket skein module, Math.
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[9] S. Tsuji, An invariant for integral homology 3‐spheres via completed Kauffman bracket
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[10] S. Tsuji, A formula for the action of Dehn twists on HOMFLY‐PT skein modules
and its applications, preprint, arXiv:1801.00580.

[11] S. Tsuji,

A construction of the HOMFLY‐PT invariant for integral homology 3‐

spheres via the HOMFLY‐PT skein algebra, in preparation.

[12] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci.
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Research Institute for Mathematical Sciences

Kyoto University
Kyoto 606‐8502
JAPAN

E‐mail address: tsujish@kurims.kyoto‐u.ac.jp