Definition 1 .

The Homflypt skein module . Let M 𝑀 M italic_M be an oriented 3 3 3 3 -manifold. The Homflypt skein module of M 𝑀 M italic_M , denoted by S ⁢ ( M ) 𝑆 𝑀 S(M) italic_S ( italic_M ) , is the k 𝑘 k italic_k -module freely generated by isotopy classes of framed oriented links in M 𝑀 M italic_M including the empty link, quotiented by the Homflypt skein relations given in the following figure.

x - 1 - x = ( s - s - 1 ) , superscript 𝑥 1 𝑥 𝑠 superscript 𝑠 1 x^{-1}\quad\raisebox{-8.535827pt}{\epsfbox{left.ai}}\quad-\quad x\quad% \raisebox{-8.535827pt}{\epsfbox{right.ai}}\quad=\quad(\ s-\ s^{-1})\quad% \raisebox{-8.535827pt}{\epsfbox{parra.ai}}\quad, italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT - italic_x = ( italic_s - italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ,
= ( x ⁢ v - 1 ) , 𝑥 superscript 𝑣 1 \raisebox{-8.535827pt}{\epsfbox{framel.ai}}\quad=\quad(xv^{-1})\quad\raisebox{% -8.535827pt}{\epsfbox{orline.ai}}\quad, = ( italic_x italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ,
L ⊔ = v - 1 - v s - s - 1 L . square-union 𝐿 superscript 𝑣 1 𝑣 𝑠 superscript 𝑠 1 𝐿 L\ \sqcup\raisebox{-5.690551pt}{\epsfbox{unknot.ai}}\quad=\quad{\dfrac{v^{-1}-% v}{\ s-\ s^{-1}}}\quad L\quad. italic_L ⊔ = divide start_ARG italic_v start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT - italic_v end_ARG start_ARG italic_s - italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG italic_L .
Definition 2.13 .

A linear connection on a Lie algebroid A 𝐴 A italic_A , with associated A 𝐴 A italic_A -derivative ∇ normal-∇ \nabla ∇ , is said to be compatible with the Lie algebroid structure of A 𝐴 A italic_A if there exists an A 𝐴 A italic_A -connection in T ⁢ M 𝑇 𝑀 TM italic_T italic_M , with associated A 𝐴 A italic_A -derivative ∇ ˇ normal-ˇ normal-∇ \check{\nabla} roman_ˇ start_ARG ∇ end_ARG , such that

∇ ˇ ⁢ # = # ⁢ ∇ . ˇ ∇ # # ∇ \check{\nabla}\#=\#\nabla. roman_ˇ start_ARG ∇ end_ARG # = # ∇ .
Definition .

Let 𝔤 𝔤 \mathfrak{g} fraktur_g be a solvable, non nilpotent Lie algebra. Then 𝔤 𝔤 \mathfrak{g} fraktur_g is called decomposable if

𝔤 = 𝔫 ⊕ 𝔱 𝔤 direct-sum 𝔫 𝔱 \mathfrak{g}=\mathfrak{n}\oplus\mathfrak{t} fraktur_g = fraktur_n ⊕ fraktur_t

where 𝔫 𝔫 \mathfrak{n} fraktur_n is the nilradical of 𝔤 𝔤 \mathfrak{g} fraktur_g and 𝔱 𝔱 \mathfrak{t} fraktur_t an exterior torus of derivations, i.e., an abelian subalgebra consisting of a ⁢ d 𝑎 𝑑 ad italic_a italic_d -semisimple endomorphisms.

Definition 1

An Ω Ω \Omega roman_Ω satisfying the strict positivity and Θ Θ \Theta roman_Θ -summability assumptions (2-3) is called admissible , as is its associated free Lagrangian. The canonical antilinear isomorphism f ↦ f ¯ : ℰ → ℰ ∗ normal-: maps-to normal-f normal-¯ normal-f normal-→ ℰ superscript ℰ normal-∗ f\mapsto\bar{f}:\mathcal{E}\rightarrow\mathcal{E}^{\ast} italic_f ↦ ¯ start_ARG italic_f end_ARG : caligraphic_E → caligraphic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is given by

( f ¯ , g ) = ⟨ f , g ⟩ ¯ 𝑓 𝑔 𝑓 𝑔 \left(\bar{f},g\right)=\left\langle f,g\right\rangle ( ¯ start_ARG italic_f end_ARG , italic_g ) = ⟨ italic_f , italic_g ⟩

for all g ∈ ℰ 𝑔 ℰ g\in\mathcal{E} italic_g ∈ caligraphic_E . Given a linear or antilinear operator A : ℰ → ℰ : 𝐴 → ℰ ℰ A:\mathcal{E}\rightarrow\mathcal{E} italic_A : caligraphic_E → caligraphic_E , the conjugate transformation A ¯ : ℰ ∗ → ℰ ∗ : ¯ 𝐴 → superscript ℰ ∗ superscript ℰ ∗ \bar{A}:\mathcal{E}^{\ast}\rightarrow\mathcal{E}^{\ast} ¯ start_ARG italic_A end_ARG : caligraphic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is given by

A ¯ ⁢ g ¯ = A ⁢ g ¯ . ¯ 𝐴 ¯ 𝑔 ¯ 𝐴 𝑔 \bar{A}\bar{g}=\overline{Ag}. ¯ start_ARG italic_A end_ARG ¯ start_ARG italic_g end_ARG = ¯ start_ARG italic_A italic_g end_ARG .