Definition 1 .

The Homflypt skein module \qua Let k 𝑘 k italic_k be a commutative ring containing x ± 1 , superscript đ‘„ plus-or-minus 1 x^{\pm 1}, italic_x start_POSTSUPERSCRIPT ± 1 end_POSTSUPERSCRIPT , v ± 1 , superscript 𝑣 plus-or-minus 1 v^{\pm 1}, italic_v start_POSTSUPERSCRIPT ± 1 end_POSTSUPERSCRIPT , s ± 1 , superscript 𝑠 plus-or-minus 1 s^{\pm 1}, italic_s start_POSTSUPERSCRIPT ± 1 end_POSTSUPERSCRIPT , and 1 s - s - 1 . 1 𝑠 superscript 𝑠 1 \frac{1}{s-s^{-1}}. divide start_ARG 1 end_ARG start_ARG italic_s - italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG . Let M 𝑀 M italic_M be an oriented 3 3 3 3 -manifold. The Homflypt skein module of M 𝑀 M italic_M over k , 𝑘 k, italic_k , denoted by S k ⁹ ( M ) subscript 𝑆 𝑘 𝑀 S_{k}(M) italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 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 đ‘„ 1 đ‘„ 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 ⁹ v - 1 ) , đ‘„ 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{\frac{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 .

The last relation follows from the first two in the case L 𝐿 L italic_L is nonempty.

Definition 2.2.1 \pointrait

Let A ∈ 𝒜 0 𝐮 subscript 𝒜 0 A\in{\mathcal{A}}_{0} italic_A ∈ caligraphic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (resp. A ∈ 𝒜 q 𝐮 subscript 𝒜 𝑞 A\in{\mathcal{A}}_{q} italic_A ∈ caligraphic_A start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) and f ∈ ℱ ⁹ ( A , S ) 𝑓 ℱ 𝐮 𝑆 f\in{\mathcal{F}}(A,S) italic_f ∈ caligraphic_F ( italic_A , italic_S ) . We say that f 𝑓 f italic_f is invariant if

f ⁱ ( n ⁱ a ) = f ⁱ ( a ) 𝑓 𝑛 𝑎 𝑓 𝑎 f(na)=f(a) italic_f ( italic_n italic_a ) = italic_f ( italic_a )

for all a ∈ A 𝑎 𝐮 a\in A italic_a ∈ italic_A and all n ∈ â„€ ( 0 ) 𝑛 subscript â„€ 0 n\in{\mathbb{Z}}_{(0)} italic_n ∈ blackboard_Z start_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT (resp. all n ∈ â„€ ( q ) 𝑛 subscript â„€ 𝑞 n\in{\mathbb{Z}}_{(q)} italic_n ∈ blackboard_Z start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT )

Let V 𝑉 V italic_V be a vector space over k 𝑘 k italic_k and f ∈ ℱ ⁹ ( V , S ) 𝑓 ℱ 𝑉 𝑆 f\in{\mathcal{F}}(V,S) italic_f ∈ caligraphic_F ( italic_V , italic_S ) . We say that f 𝑓 f italic_f is invariant if

f ⁹ ( Îș ⁹ v ) = f ⁹ ( v ) 𝑓 𝜅 𝑣 𝑓 𝑣 f(\kappa v)=f(v) italic_f ( italic_Îș italic_v ) = italic_f ( italic_v )

for all Îș ∈ k * 𝜅 superscript 𝑘 \kappa\in k^{*} italic_Îș ∈ italic_k start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT and v ∈ V 𝑣 𝑉 v\in V italic_v ∈ italic_V .