Definition 1.1 .

For any k ∈ β„‚ d π‘˜ superscript β„‚ 𝑑 k\in\mathbb{C}^{d} italic_k ∈ blackboard_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , the subspace H k s = H k s ⁒ ( W ) βŠ‚ H s ⁒ ( W ) subscript superscript 𝐻 𝑠 π‘˜ subscript superscript 𝐻 𝑠 π‘˜ π‘Š superscript 𝐻 𝑠 π‘Š H^{s}_{k}=H^{s}_{k}(W)\subset H^{s}(W) italic_H start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_W ) βŠ‚ italic_H start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_W ) consists of the restrictions to W π‘Š W italic_W of functions f ∈ H π‘™π‘œπ‘ s ⁒ ( ℝ d ) 𝑓 subscript superscript 𝐻 𝑠 π‘™π‘œπ‘ superscript ℝ 𝑑 f\in H^{s}_{\text{loc}}(\mathbb{R}^{d}) italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT loc end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) that satisfy the Floquet-Bloch condition that for any Ξ³ ∈ β„€ d 𝛾 superscript β„€ 𝑑 \gamma\in\mathbb{Z}^{d} italic_Ξ³ ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT

f ⁒ ( x + Ξ³ ) = e i ⁒ k β‹… Ξ³ ⁒ f ⁒ ( x ) ⁒ a.e. 𝑓 π‘₯ 𝛾 superscript 𝑒 β‹… 𝑖 π‘˜ 𝛾 𝑓 π‘₯ a.e. f(x+\gamma)=e^{ik\cdot\gamma}f(x)\ \text{a.e.} italic_f ( italic_x + italic_Ξ³ ) = italic_e start_POSTSUPERSCRIPT italic_i italic_k β‹… italic_Ξ³ end_POSTSUPERSCRIPT italic_f ( italic_x ) a.e. (4)

Here H s superscript 𝐻 𝑠 H^{s} italic_H start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT denotes the standard Sobolev space of order s 𝑠 s italic_s .

Definition 19 (Reeb Graph)

Let f : ℝ 2 β†’ ℝ normal-: 𝑓 normal-β†’ superscript ℝ 2 ℝ f:\mathbb{R}^{2}\to\mathbb{R} italic_f : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT β†’ blackboard_R be a function. Define a equivalence relation ∼ similar-to \sim ∼ on the space πΊπ‘Ÿπ‘Žπ‘β„Ž ⁒ ( f ) ≐ { ( x , f ⁒ ( x ) ) : x ∈ ℝ 2 } approaches-limit πΊπ‘Ÿπ‘Žπ‘β„Ž 𝑓 conditional-set π‘₯ 𝑓 π‘₯ π‘₯ superscript ℝ 2 \mbox{Graph}(f)\doteq\{(x,f(x)):\,x\in\mathbb{R}^{2}\} Graph ( italic_f ) ≐ { ( italic_x , italic_f ( italic_x ) ) : italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT } by

( x , f ⁒ ( x ) ) ∼ ( y , f ⁒ ( y ) ) ⁒ 𝑖𝑓𝑓 ⁒ f ⁒ ( x ) = f ⁒ ( y ) ⁒ and there is a continuous path from x to y in f - 1 ⁒ ( f ⁒ ( x ) ) . similar-to π‘₯ 𝑓 π‘₯ 𝑦 𝑓 𝑦 𝑖𝑓𝑓 𝑓 π‘₯ 𝑓 𝑦 and there is a continuous path from x to y in f - 1 ⁒ ( f ⁒ ( x ) ) (x,f(x))\sim(y,f(y))\,\mbox{iff}\,f(x)=f(y)\,\mbox{and there is a continuous path from $x$ to $y$ in $f^{-1}(f(x))$}. ( italic_x , italic_f ( italic_x ) ) ∼ ( italic_y , italic_f ( italic_y ) ) iff italic_f ( italic_x ) = italic_f ( italic_y ) and there is a continuous path from italic_x to italic_y in italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_f ( italic_x ) ) .

The Reeb graph of the function f 𝑓 f italic_f , denoted R ⁒ e ⁒ e ⁒ b ⁒ ( f ) 𝑅 𝑒 𝑒 𝑏 𝑓 Reeb(f) italic_R italic_e italic_e italic_b ( italic_f ) , is the topological space πΊπ‘Ÿπ‘Žπ‘β„Ž ( f ) / ∼ fragments πΊπ‘Ÿπ‘Žπ‘β„Ž fragments normal-( f normal-) similar-to \mbox{Graph}(f)/\sim Graph ( italic_f ) / ∼ .

Definition 2.13 .

A computable ΞΌ πœ‡ \mu italic_ΞΌ -martingale is a computable function d : 2 < Ο‰ β†’ ℝ β‰₯ 0 normal-: 𝑑 normal-β†’ superscript 2 absent πœ” superscript ℝ absent 0 d:2^{<\omega}\rightarrow\mathbb{R}^{\geq 0} italic_d : 2 start_POSTSUPERSCRIPT < italic_Ο‰ end_POSTSUPERSCRIPT β†’ blackboard_R start_POSTSUPERSCRIPT β‰₯ 0 end_POSTSUPERSCRIPT such that for every Οƒ ∈ 2 < Ο‰ 𝜎 superscript 2 absent πœ” \sigma\in 2^{<\omega} italic_Οƒ ∈ 2 start_POSTSUPERSCRIPT < italic_Ο‰ end_POSTSUPERSCRIPT ,

ΞΌ ⁒ ( Οƒ ) ⁒ d ⁒ ( Οƒ ) = ΞΌ ⁒ ( Οƒ ⁒ 0 ) ⁒ d ⁒ ( Οƒ ⁒ 0 ) + ΞΌ ⁒ ( Οƒ ⁒ 1 ) ⁒ d ⁒ ( Οƒ ⁒ 1 ) . πœ‡ 𝜎 𝑑 𝜎 πœ‡ 𝜎 0 𝑑 𝜎 0 πœ‡ 𝜎 1 𝑑 𝜎 1 \mu(\sigma)d(\sigma)=\mu(\sigma 0)d(\sigma 0)+\mu(\sigma 1)d(\sigma 1). italic_ΞΌ ( italic_Οƒ ) italic_d ( italic_Οƒ ) = italic_ΞΌ ( italic_Οƒ 0 ) italic_d ( italic_Οƒ 0 ) + italic_ΞΌ ( italic_Οƒ 1 ) italic_d ( italic_Οƒ 1 ) .

Moreover, a computable ΞΌ πœ‡ \mu italic_ΞΌ -martingale d 𝑑 d italic_d succeeds on x ∈ 2 Ο‰ π‘₯ superscript 2 πœ” x\in 2^{\omega} italic_x ∈ 2 start_POSTSUPERSCRIPT italic_Ο‰ end_POSTSUPERSCRIPT if

lim sup n β†’ ∞ d ( x β†Ύ n ) = ∞ . fragments subscript limit-supremum β†’ 𝑛 d fragments ( x β†Ύ n ) . \limsup_{n\rightarrow\infty}d(x{\upharpoonright}n)=\infty. lim sup start_POSTSUBSCRIPT italic_n β†’ ∞ end_POSTSUBSCRIPT italic_d ( italic_x β†Ύ italic_n ) = ∞ .
Definition 3.1

For any ( p , q ) ∈ I ⁒ R 2 𝑝 π‘ž normal-I superscript normal-R 2 (p,q)\in{\rm I}\!{\rm R}^{2} ( italic_p , italic_q ) ∈ roman_I roman_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , we define the paraboloid

w ⁒ ( p , q , r ) ⁒ ( s ) = u ⁒ ( r ) + p ⁒ ( s - r ) + q 2 ⁒ ( s - r ) 2 . 𝑀 𝑝 π‘ž π‘Ÿ 𝑠 𝑒 π‘Ÿ 𝑝 𝑠 π‘Ÿ π‘ž 2 superscript 𝑠 π‘Ÿ 2 w(p,q,r)(s)=u(r)+p(s-r)+{q\over 2}(s-r)^{2}. italic_w ( italic_p , italic_q , italic_r ) ( italic_s ) = italic_u ( italic_r ) + italic_p ( italic_s - italic_r ) + divide start_ARG italic_q end_ARG start_ARG 2 end_ARG ( italic_s - italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .
Definition 2.5 (Grading rule) .

Let π’ž π’ž \mathcal{C} caligraphic_C be a k π‘˜ k italic_k -dimensional configuration, a ∈ V 0 ⁒ ( π’ž ) π‘Ž subscript 𝑉 0 π’ž a\in V_{0}(\mathcal{C}) italic_a ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_C ) and b ∈ V 1 ⁒ ( π’ž ) 𝑏 subscript 𝑉 1 π’ž b\in V_{1}(\mathcal{C}) italic_b ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( caligraphic_C ) monomials. If F C ⁒ ( a , b ) = 1 subscript 𝐹 𝐢 π‘Ž 𝑏 1 F_{C}(a,b)=1 italic_F start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_a , italic_b ) = 1 implies

g ⁒ r ⁒ ( b ) - g ⁒ r ⁒ ( a ) = k - 2 , 𝑔 π‘Ÿ 𝑏 𝑔 π‘Ÿ π‘Ž π‘˜ 2 gr(b)-gr(a)=k-2, italic_g italic_r ( italic_b ) - italic_g italic_r ( italic_a ) = italic_k - 2 ,

then we say F π’ž subscript 𝐹 π’ž F_{\mathcal{C}} italic_F start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT satisfies the grading rule.

Definition 2.12 (Section 1 of [ Haa79 ] ) .

The extended positive cone of M 𝑀 M italic_M , denoted M + ^ ^ superscript 𝑀 \widehat{M^{+}} ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG , is the set of weights on the predual of M 𝑀 M italic_M , i.e., maps m : M * + β†’ [ 0 , ∞ ] : π‘š β†’ superscript subscript 𝑀 0 m\colon M_{*}^{+}\to[0,\infty] italic_m : italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT β†’ [ 0 , ∞ ] such that

m ⁒ ( Ξ» ⁒ Ο• + ψ ) = Ξ» ⁒ m ⁒ ( Ο• ) + m ⁒ ( ψ ) π‘š πœ† italic-Ο• πœ“ πœ† π‘š italic-Ο• π‘š πœ“ m(\lambda\phi+\psi)=\lambda m(\phi)+m(\psi) italic_m ( italic_Ξ» italic_Ο• + italic_ψ ) = italic_Ξ» italic_m ( italic_Ο• ) + italic_m ( italic_ψ ) for all Ξ» β‰₯ 0 πœ† 0 \lambda\geq 0 italic_Ξ» β‰₯ 0 and Ο• , ψ ∈ M * + italic-Ο• πœ“ superscript subscript 𝑀 \phi,\psi\in M_{*}^{+} italic_Ο• , italic_ψ ∈ italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , and

m π‘š m italic_m is lower semicontinuous.

The extended positive cone has additional structure:

There is a natural inclusion M + β†’ M + ^ β†’ superscript 𝑀 ^ superscript 𝑀 M^{+}\to\widehat{M^{+}} italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT β†’ ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG by m ↦ ( Ο• ↦ Ο• ( m ) ) fragments m maps-to fragments ( Ο• maps-to Ο• fragments ( m ) ) m\mapsto(\phi\mapsto\phi(m)) italic_m ↦ ( italic_Ο• ↦ italic_Ο• ( italic_m ) ) .

For m ∈ M + ^ π‘š ^ superscript 𝑀 m\in\widehat{M^{+}} italic_m ∈ ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG and a ∈ M π‘Ž 𝑀 a\in M italic_a ∈ italic_M , we define a * ⁒ m ⁒ a ∈ M + ^ superscript π‘Ž π‘š π‘Ž ^ superscript 𝑀 a^{*}ma\in\widehat{M^{+}} italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_m italic_a ∈ ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG by

a * ⁒ m ⁒ a ⁒ ( Ο• ) = m ⁒ ( a ⁒ Ο• ⁒ a * ) = m ⁒ ( Ο• ⁒ ( a * β‹… a ) ) . superscript π‘Ž π‘š π‘Ž italic-Ο• π‘š π‘Ž italic-Ο• superscript π‘Ž π‘š italic-Ο• β‹… superscript π‘Ž π‘Ž a^{*}ma(\phi)=m(a\phi a^{*})=m(\phi(a^{*}\,\cdot\,a)). italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_m italic_a ( italic_Ο• ) = italic_m ( italic_a italic_Ο• italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) = italic_m ( italic_Ο• ( italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT β‹… italic_a ) ) .

We write Ξ» ⁒ m πœ† π‘š \lambda m italic_Ξ» italic_m for Ξ» 1 / 2 ⁒ m ⁒ Ξ» 1 / 2 superscript πœ† 1 2 π‘š superscript πœ† 1 2 \lambda^{1/2}m\lambda^{1/2} italic_Ξ» start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT italic_m italic_Ξ» start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT for Ξ» β‰₯ 0 πœ† 0 \lambda\geq 0 italic_Ξ» β‰₯ 0 .

There is a natural partial ordering on M + ^ ^ superscript 𝑀 \widehat{M^{+}} ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG given by m 1 ≀ m 2 subscript π‘š 1 subscript π‘š 2 m_{1}\leq m_{2} italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≀ italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if m 1 ⁒ ( Ο• ) ≀ m 2 ⁒ ( Ο• ) subscript π‘š 1 italic-Ο• subscript π‘š 2 italic-Ο• m_{1}(\phi)\leq m_{2}(\phi) italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_Ο• ) ≀ italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_Ο• ) for all Ο• ∈ M * + italic-Ο• superscript subscript 𝑀 \phi\in M_{*}^{+} italic_Ο• ∈ italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT .

If I 𝐼 I italic_I is a directed set, we say ( m i ) i ∈ I βŠ‚ M + ^ subscript subscript π‘š 𝑖 𝑖 𝐼 ^ superscript 𝑀 (m_{i})_{i\in I}\subset\widehat{M^{+}} ( italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT βŠ‚ ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG increases to m ∈ M + ^ π‘š ^ superscript 𝑀 m\in\widehat{M^{+}} italic_m ∈ ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG if i ≀ j 𝑖 𝑗 i\leq j italic_i ≀ italic_j implies m i ≀ m j subscript π‘š 𝑖 subscript π‘š 𝑗 m_{i}\leq m_{j} italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≀ italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and sup i ⁑ m i ⁒ ( Ο• ) = m ⁒ ( Ο• ) subscript supremum 𝑖 subscript π‘š 𝑖 italic-Ο• π‘š italic-Ο• \sup_{i}m_{i}(\phi)=m(\phi) roman_sup start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_Ο• ) = italic_m ( italic_Ο• ) for all Ο• ∈ M * + italic-Ο• superscript subscript 𝑀 \phi\in M_{*}^{+} italic_Ο• ∈ italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT . Hence we can define the sum of elements of M + ^ ^ superscript 𝑀 \widehat{M^{+}} ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG pointwise.

Each Ο• ∈ M * + italic-Ο• superscript subscript 𝑀 \phi\in M_{*}^{+} italic_Ο• ∈ italic_M start_POSTSUBSCRIPT * end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT extends uniquely to a map M + ^ β†’ [ 0 , ∞ ] β†’ ^ superscript 𝑀 0 \widehat{M^{+}}\to[0,\infty] ^ start_ARG italic_M start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_ARG β†’ [ 0 , ∞ ] by Ο• ⁒ ( m ) = m ⁒ ( Ο• ) italic-Ο• π‘š π‘š italic-Ο• \phi(m)=m(\phi) italic_Ο• ( italic_m ) = italic_m ( italic_Ο• ) .

Definition \thedefs .

A point ( x , y , z ) ∈ π•Š n Γ— π•Š m Γ— π•Š s π‘₯ 𝑦 𝑧 superscript π•Š 𝑛 superscript π•Š π‘š superscript π•Š 𝑠 (x,y,z)\in\mathbb{S}^{n}\times\mathbb{S}^{m}\times\mathbb{S}^{s} ( italic_x , italic_y , italic_z ) ∈ blackboard_S start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT Γ— blackboard_S start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT Γ— blackboard_S start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT is called an extreme point of β„“ normal-β„“ \ell roman_β„“ if it satisfies the system of equations

βˆ‡ ⁑ β„“ ⁒ ( x , y , z ) = ( 2 ⁒ Ξ± ⁒ x , 2 ⁒ Ξ² ⁒ y , 2 ⁒ Ξ» ⁒ z ) , βˆ‡ β„“ π‘₯ 𝑦 𝑧 2 𝛼 π‘₯ 2 𝛽 𝑦 2 πœ† 𝑧 \nabla\ell(x,y,z)=(2\alpha x,2\beta y,2\lambda z), βˆ‡ roman_β„“ ( italic_x , italic_y , italic_z ) = ( 2 italic_Ξ± italic_x , 2 italic_Ξ² italic_y , 2 italic_Ξ» italic_z ) ,

for some Ξ± , Ξ² , Ξ» ∈ ℝ 𝛼 𝛽 πœ† ℝ \alpha,\beta,\lambda\in\mathbb{R} italic_Ξ± , italic_Ξ² , italic_Ξ» ∈ blackboard_R . Note that if ( x , y , z ) π‘₯ 𝑦 𝑧 (x,y,z) ( italic_x , italic_y , italic_z ) is an extreme point, then ( Β± x , Β± y , Β± z ) plus-or-minus π‘₯ plus-or-minus 𝑦 plus-or-minus 𝑧 (\pm x,\pm y,\pm z) ( Β± italic_x , Β± italic_y , Β± italic_z ) is also an extreme point. We say that they belong to the same class .

Definition 2.4

Let π’œ π’œ \mathcal{A} caligraphic_A and π’ž π’ž \mathcal{C} caligraphic_C be associative algebras. An algebra homomorphism is a map Ο† : π’œ β†’ π’ž normal-: πœ‘ normal-β†’ π’œ π’ž \varphi:\mathcal{A}\to\mathcal{C} italic_Ο† : caligraphic_A β†’ caligraphic_C such that for all x , y ∈ π’œ π‘₯ 𝑦 π’œ x,y\in\mathcal{A} italic_x , italic_y ∈ caligraphic_A ,

Ο† ⁒ ( x ⁒ y ) = Ο† ⁒ ( x ) ⁒ Ο† ⁒ ( y ) . πœ‘ π‘₯ 𝑦 πœ‘ π‘₯ πœ‘ 𝑦 \varphi(xy)=\varphi(x)\varphi(y). italic_Ο† ( italic_x italic_y ) = italic_Ο† ( italic_x ) italic_Ο† ( italic_y ) . (1)
Definition 4.3 .

A twisting morphism is an element Ξ± ∈ 𝐇𝐨𝐦 ⁒ ( π’ž , 𝒫 ) 𝛼 𝐇𝐨𝐦 π’ž 𝒫 \alpha\in{\textbf{Hom}}(\mathcal{C},\mathcal{P}) italic_Ξ± ∈ Hom ( caligraphic_C , caligraphic_P ) of degree 1 1 1 1 satisfying the Maurer-Cartan equation

βˆ‚ ⁑ ( Ξ± ) + Ξ± ⋆ Ξ± = 0 . 𝛼 ⋆ 𝛼 𝛼 0 \partial(\alpha)+\alpha\star\alpha=0. βˆ‚ ( italic_Ξ± ) + italic_Ξ± ⋆ italic_Ξ± = 0 .

We denote the set of twisting morphisms by Tw ⁒ ( π’ž , 𝒫 ) Tw π’ž 𝒫 {\rm Tw}(\mathcal{C},\mathcal{P}) roman_Tw ( caligraphic_C , caligraphic_P ) .

Definition 2

Let L 𝐿 L italic_L be an oriented virtual knot or link diagram and let X = { x 1 , … , x n } 𝑋 subscript π‘₯ 1 … subscript π‘₯ 𝑛 X=\{x_{1},\dots,x_{n}\} italic_X = { italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } be a set of generators corresponding to the semiarcs of L 𝐿 L italic_L . Then the Alexander biquandle of L 𝐿 L italic_L , denoted A ⁒ B ⁒ ( L ) 𝐴 𝐡 𝐿 AB(L) italic_A italic_B ( italic_L ) , is the Ξ› Ξ› \Lambda roman_Ξ› -module generated by X 𝑋 X italic_X with the relations pictured below at positively ( + + + ) and negatively ( - - - ) oriented classical crossings and at virtual crossings:

z = t ⁒ y + ( 1 - s ⁒ t ) ⁒ x w = s ⁒ x z = y w = x 𝑧 𝑑 𝑦 1 𝑠 𝑑 π‘₯ 𝑀 𝑠 π‘₯ 𝑧 𝑦 𝑀 π‘₯ \begin{array}[]{cc}\quad\quad\includegraphics[]{ac-ah-sn-1.png}&\quad\quad% \includegraphics[]{ac-ah-sn-2.png}\\ \begin{array}[]{rcl}z&=&ty+(1-st)x\\ w&=&sx\end{array}&\begin{array}[]{rcl}z&=&y\\ w&=&x\end{array}\end{array} start_ARRAY start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL start_ARRAY start_ROW start_CELL italic_z end_CELL start_CELL = end_CELL start_CELL italic_t italic_y + ( 1 - italic_s italic_t ) italic_x end_CELL end_ROW start_ROW start_CELL italic_w end_CELL start_CELL = end_CELL start_CELL italic_s italic_x end_CELL end_ROW end_ARRAY end_CELL start_CELL start_ARRAY start_ROW start_CELL italic_z end_CELL start_CELL = end_CELL start_CELL italic_y end_CELL end_ROW start_ROW start_CELL italic_w end_CELL start_CELL = end_CELL start_CELL italic_x end_CELL end_ROW end_ARRAY end_CELL end_ROW end_ARRAY

In particular, A ⁒ B ⁒ ( L ) 𝐴 𝐡 𝐿 AB(L) italic_A italic_B ( italic_L ) is obtained from the fundamental virtual biquandle of L 𝐿 L italic_L by setting

B ⁒ ( x , y ) = ( t ⁒ y + ( 1 - s ⁒ t ) ⁒ x , s ⁒ x ) and V ⁒ ( x , y ) = ( y , x ) . formulae-sequence 𝐡 π‘₯ 𝑦 𝑑 𝑦 1 𝑠 𝑑 π‘₯ 𝑠 π‘₯ and 𝑉 π‘₯ 𝑦 𝑦 π‘₯ B(x,y)=(ty+(1-st)x,sx)\quad\mathrm{and}\quad V(x,y)=(y,x). italic_B ( italic_x , italic_y ) = ( italic_t italic_y + ( 1 - italic_s italic_t ) italic_x , italic_s italic_x ) roman_and italic_V ( italic_x , italic_y ) = ( italic_y , italic_x ) .
Definition 2.2 .

Given z ∈ 𝕋 N 𝑧 subscript 𝕋 𝑁 z\in\mathbb{T}_{N} italic_z ∈ blackboard_T start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and a word b ∈ ( 0 , ∞ ) N 𝑏 superscript 0 𝑁 b\in(0,\infty)^{N} italic_b ∈ ( 0 , ∞ ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT . Geometric row insertion of b 𝑏 b italic_b into z 𝑧 z italic_z outputs a new triangular array z β€² ∈ 𝕋 N superscript 𝑧 β€² subscript 𝕋 𝑁 z^{\prime}\in\mathbb{T}_{N} italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∈ blackboard_T start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT . This procedure is denoted by

(2.3) z β€² = z ← b superscript 𝑧 β€² 𝑧 ← 𝑏 z^{\prime}=z\leftarrow b italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT = italic_z ← italic_b

and it consists of N 𝑁 N italic_N iterations of the basic row insertion. For 1 ≀ β„“ ≀ N 1 β„“ 𝑁 1\leq\ell\leq N 1 ≀ roman_β„“ ≀ italic_N form words z β„“ = ( z β„“ ⁒ β„“ , … , z N ⁒ β„“ ) subscript 𝑧 β„“ subscript 𝑧 β„“ β„“ … subscript 𝑧 𝑁 β„“ z_{\ell}=(z_{\ell\ell},\ldots,z_{N\ell}) italic_z start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT = ( italic_z start_POSTSUBSCRIPT roman_β„“ roman_β„“ end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_N roman_β„“ end_POSTSUBSCRIPT ) . Begin by setting a 1 = b subscript π‘Ž 1 𝑏 a_{1}=b italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_b . Then for β„“ = 1 , … , N β„“ 1 … 𝑁 \ell=1,\ldots,N roman_β„“ = 1 , … , italic_N recursively apply the map

(2.4) a β„“ z β„“ ⟢ ↓ z β„“ β€² a β„“ + 1 missing-subexpression subscript π‘Ž β„“ missing-subexpression subscript 𝑧 β„“ ⟢ ↓ subscript superscript 𝑧 β€² β„“ missing-subexpression subscript π‘Ž β„“ 1 missing-subexpression \begin{array}[]{ccc}&a_{\ell}&\\ z_{\ell}&\begin{picture}(12.0,10.0)(-2.0,0.0)\put(-8.0,0.0){\Large$% \longrightarrow$} \put(0.0,0.0){\Large$\downarrow$} \end{picture}&z^{\prime}_{\ell}\\ &a_{\ell+1}\end{array} start_ARRAY start_ROW start_CELL end_CELL start_CELL italic_a start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_z start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT end_CELL start_CELL ⟢ ↓ end_CELL start_CELL italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_a start_POSTSUBSCRIPT roman_β„“ + 1 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY

from Definition 2.1 , where a β„“ + 1 = a β„“ β€² subscript π‘Ž β„“ 1 subscript superscript π‘Ž β€² β„“ a_{\ell+1}=a^{\prime}_{\ell} italic_a start_POSTSUBSCRIPT roman_β„“ + 1 end_POSTSUBSCRIPT = italic_a start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT . The last output a N + 1 subscript π‘Ž 𝑁 1 a_{N+1} italic_a start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT is empty. The new array z β€² = ( z k ⁒ β„“ β€² : 1 ≀ β„“ ≀ k ≀ N ) fragments superscript 𝑧 β€² fragments ( subscript superscript 𝑧 β€² π‘˜ β„“ : 1 β„“ k N ) z^{\prime}=(z^{\prime}_{k\ell}:1\leq\ell\leq k\leq N) italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT = ( italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k roman_β„“ end_POSTSUBSCRIPT : 1 ≀ roman_β„“ ≀ italic_k ≀ italic_N ) is formed from the words z β„“ β€² = ( z β„“ ⁒ β„“ β€² , … , z N ⁒ β„“ β€² ) subscript superscript 𝑧 β€² β„“ subscript superscript 𝑧 β€² β„“ β„“ … subscript superscript 𝑧 β€² 𝑁 β„“ z^{\prime}_{\ell}=(z^{\prime}_{\ell\ell},\ldots,z^{\prime}_{N\ell}) italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT = ( italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_β„“ roman_β„“ end_POSTSUBSCRIPT , … , italic_z start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N roman_β„“ end_POSTSUBSCRIPT ) . Along the way the procedure constructs an auxiliary triangular array a = ( a k ⁒ β„“ : 1 ≀ β„“ ≀ k ≀ N ) fragments a fragments ( subscript π‘Ž π‘˜ β„“ : 1 β„“ k N ) a=(a_{k\ell}:1\leq\ell\leq k\leq N) italic_a = ( italic_a start_POSTSUBSCRIPT italic_k roman_β„“ end_POSTSUBSCRIPT : 1 ≀ roman_β„“ ≀ italic_k ≀ italic_N ) with diagonals a β„“ = ( a β„“ ⁒ β„“ , … , a N ⁒ β„“ ) subscript π‘Ž β„“ subscript π‘Ž β„“ β„“ … subscript π‘Ž 𝑁 β„“ a_{\ell}=(a_{\ell\ell},\dotsc,a_{N\ell}) italic_a start_POSTSUBSCRIPT roman_β„“ end_POSTSUBSCRIPT = ( italic_a start_POSTSUBSCRIPT roman_β„“ roman_β„“ end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_N roman_β„“ end_POSTSUBSCRIPT ) .

Definition 2.10 ( 𝖱𝖒𝖠 0 subscript 𝖱𝖒𝖠 0 \mathsf{RCA}_{0} sansserif_RCA start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

Suppose ( X , 𝒰 , k ) 𝑋 𝒰 π‘˜ (X,\mathcal{U},k) ( italic_X , caligraphic_U , italic_k ) and ( Y , 𝒱 , β„“ ) π‘Œ 𝒱 β„“ (Y,\mathcal{V},\ell) ( italic_Y , caligraphic_V , roman_β„“ ) are countable second-countable space with 𝒰 = ⟨ U i ⟩ i ∈ I 𝒰 subscript delimited-⟨⟩ subscript π‘ˆ 𝑖 𝑖 𝐼 \mathcal{U}=\langle U_{i}\rangle_{i\in I} caligraphic_U = ⟨ italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT and 𝒱 = ⟨ V j ⟩ j ∈ J . 𝒱 subscript delimited-⟨⟩ subscript 𝑉 𝑗 𝑗 𝐽 \mathcal{V}=\langle V_{j}\rangle_{j\in J}. caligraphic_V = ⟨ italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_j ∈ italic_J end_POSTSUBSCRIPT . The product space ( X , 𝒰 , k ) Γ— ( Y , 𝒱 , β„“ ) 𝑋 𝒰 π‘˜ π‘Œ 𝒱 β„“ (X,\mathcal{U},k)\times(Y,\mathcal{V},\ell) ( italic_X , caligraphic_U , italic_k ) Γ— ( italic_Y , caligraphic_V , roman_β„“ ) is the countable second-countable space ( X Γ— Y , 𝒲 , m ) , 𝑋 π‘Œ 𝒲 π‘š (X\times Y,\mathcal{W},m), ( italic_X Γ— italic_Y , caligraphic_W , italic_m ) , where 𝒲 = ⟨ U i Γ— V j ⟩ ⟨ i , j ⟩ ∈ I Γ— J 𝒲 subscript delimited-⟨⟩ subscript π‘ˆ 𝑖 subscript 𝑉 𝑗 𝑖 𝑗 𝐼 𝐽 \mathcal{W}=\langle U_{i}\times V_{j}\rangle_{\langle i,j\rangle\in I\times J} caligraphic_W = ⟨ italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT Γ— italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT ⟨ italic_i , italic_j ⟩ ∈ italic_I Γ— italic_J end_POSTSUBSCRIPT and

m ⁒ ( ⟨ x , y ⟩ , ⟨ i , j ⟩ , ⟨ i β€² , j β€² ⟩ ) = ⟨ k ⁒ ( x , i , i β€² ) , β„“ ⁒ ( y , j , j β€² ) ⟩ . π‘š π‘₯ 𝑦 𝑖 𝑗 superscript 𝑖 β€² superscript 𝑗 β€² π‘˜ π‘₯ 𝑖 superscript 𝑖 β€² β„“ 𝑦 𝑗 superscript 𝑗 β€² m(\langle x,y\rangle,\langle i,j\rangle,\langle i^{\prime},j^{\prime}\rangle)=% \langle k(x,i,i^{\prime}),\ell(y,j,j^{\prime})\rangle. italic_m ( ⟨ italic_x , italic_y ⟩ , ⟨ italic_i , italic_j ⟩ , ⟨ italic_i start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ⟩ ) = ⟨ italic_k ( italic_x , italic_i , italic_i start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) , roman_β„“ ( italic_y , italic_j , italic_j start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) ⟩ .
Definition 1.1 .

The reproductive equations are the equations of the following form:

x = Ο† ⁒ ( x ) , π‘₯ πœ‘ π‘₯ x=\varphi(x), italic_x = italic_Ο† ( italic_x ) ,

where x π‘₯ x italic_x is a unknown, S 𝑆 S italic_S is a given set and Ο† : S ⟢ S : πœ‘ ⟢ 𝑆 𝑆 \varphi:S\longrightarrow S italic_Ο† : italic_S ⟢ italic_S is a given function which satisfies the following condition:

(1.1) Ο† ∘ Ο† = Ο† . πœ‘ πœ‘ πœ‘ \varphi\circ\varphi=\varphi. italic_Ο† ∘ italic_Ο† = italic_Ο† .
Definition 4.7 (Tameness) .

Let ( M 4 , Ξ£ ) superscript 𝑀 4 normal-Ξ£ (M^{4},\Sigma) ( italic_M start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , roman_Ξ£ ) be a 4 4 4 4 -dimensional manifold with a codimension- 2 2 2 2 submanifold with isolated singularities. We say that a 1 1 1 1 -form Ξ» πœ† \lambda italic_Ξ» is tame at U βŠ‚ Ξ£ Reg π‘ˆ superscript normal-Ξ£ Reg U\subset\Sigma^{\textnormal{\tiny Reg}} italic_U βŠ‚ roman_Ξ£ start_POSTSUPERSCRIPT Reg end_POSTSUPERSCRIPT if on some neighbourhood V 𝑉 V italic_V of U π‘ˆ U italic_U in M 𝑀 M italic_M , there exists a smooth function ΞΊ πœ… \kappa italic_ΞΊ and a smooth 1 1 1 1 -form ΞΌ πœ‡ \mu italic_ΞΌ , both with bounded derivatives, such that

Ξ» = ΞΊ ⁒ Ξ± + ΞΌ , πœ† πœ… 𝛼 πœ‡ \lambda=\kappa\alpha+\mu, italic_Ξ» = italic_ΞΊ italic_Ξ± + italic_ΞΌ ,

where Ξ± 𝛼 \alpha italic_Ξ± is an angular form over Ξ£ Reg superscript normal-Ξ£ Reg \Sigma^{\textnormal{\tiny Reg}} roman_Ξ£ start_POSTSUPERSCRIPT Reg end_POSTSUPERSCRIPT .

Definition 2.18 (Chapoton [ 3 ] ) .

The permutation operad, 𝒫 ⁒ e ⁒ r ⁒ m 𝒫 𝑒 π‘Ÿ π‘š \mathcal{P}erm caligraphic_P italic_e italic_r italic_m , is a binary quadratic operad over ( 1 β‹„ 2 , 2 β‹„ 1 ) normal-β‹„ 1 2 normal-β‹„ 2 1 (1\diamond 2,2\diamond 1) ( 1 β‹„ 2 , 2 β‹„ 1 ) with the quadratic relation,

( 1 β‹„ 2 ) β‹„ 3 = ( 2 β‹„ 1 ) β‹„ 3 = 1 β‹„ ( 2 β‹„ 3 ) . β‹„ β‹„ 1 2 3 β‹„ β‹„ 2 1 3 β‹„ 1 β‹„ 2 3 (1\diamond 2)\diamond 3=(2\diamond 1)\diamond 3=1\diamond(2\diamond 3). ( 1 β‹„ 2 ) β‹„ 3 = ( 2 β‹„ 1 ) β‹„ 3 = 1 β‹„ ( 2 β‹„ 3 ) .
Definition 9.14 .

For all 𝐳 𝐳 {\bf z} bold_z , 𝐳 𝐳 {\bf z} bold_z on a pair of bits in the x π‘₯ x italic_x or x β€² superscript π‘₯ normal-β€² x^{\prime} italic_x start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT binary, returns the same bits when and-or applied, where this applies to all remaining pairwise combinations, or

𝐳 ( 0 ∧ ∨ 0 ) = 00 , 𝐳 ( 1 ∧ ∨ 1 ) = 11 , 𝐳 ( 0 ∧ ∨ 1 ) = 01 , 𝐳 ( 1 ∧ ∨ 0 ) = 10 . fragments z fragments ( 0 0 ) 00 , z fragments ( 1 1 ) 11 , z fragments ( 0 1 ) 01 , z fragments ( 1 0 ) 10 . \mathbf{z}(0\wedge\vee 0)=00,\ \mathbf{z}(1\wedge\vee 1)=11,\ \mathbf{z}(0% \wedge\vee 1)=01,\ \mathbf{z}(1\wedge\vee 0)=10. bold_z ( 0 ∧ ∨ 0 ) = 00 , bold_z ( 1 ∧ ∨ 1 ) = 11 , bold_z ( 0 ∧ ∨ 1 ) = 01 , bold_z ( 1 ∧ ∨ 0 ) = 10 . (14a)
Definition 9.18 .

For all i , bit pairs in the binary of x π‘₯ x italic_x or x β€² superscript π‘₯ normal-β€² x^{\prime} italic_x start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT is either 01 or 10. i closes with 1 for 01, and 0 for 10, or

𝐒 ⁒ ( 01 ) = 0 ⟢ β†· 1 = 1 , 𝐒 ⁒ ( 10 ) = 1 ⟢ β†· 0 = 0 . formulae-sequence 𝐒 01 0 superscript ⟢ β†· 1 1 𝐒 10 1 superscript ⟢ β†· 0 0 \mathbf{i}(01)=0\stackrel{{\scriptstyle\curvearrowright}}{{\longrightarrow}}1=% 1,\ \ \ \mathbf{i}(10)=1\stackrel{{\scriptstyle\curvearrowright}}{{% \longrightarrow}}0=0. bold_i ( 01 ) = 0 start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG β†· end_ARG end_RELOP 1 = 1 , bold_i ( 10 ) = 1 start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG β†· end_ARG end_RELOP 0 = 0 . (15a)
Definition 9.19 .

For all p , bit pairs in the binary of x π‘₯ x italic_x or x β€² superscript π‘₯ normal-β€² x^{\prime} italic_x start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT is either 00 or 11. p closes with 1 for 11, and 0 for 00, or

𝐩 ⁒ ( 11 ) = 1 ⟢ β†· 1 = 1 , 𝐩 ⁒ ( 00 ) = 0 ⟢ β†· 0 = 0 . formulae-sequence 𝐩 11 1 superscript ⟢ β†· 1 1 𝐩 00 0 superscript ⟢ β†· 0 0 \mathbf{p}(11)=1\stackrel{{\scriptstyle\curvearrowright}}{{\longrightarrow}}1=% 1,\ \ \ \mathbf{p}(00)=0\stackrel{{\scriptstyle\curvearrowright}}{{% \longrightarrow}}0=0. bold_p ( 11 ) = 1 start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG β†· end_ARG end_RELOP 1 = 1 , bold_p ( 00 ) = 0 start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG β†· end_ARG end_RELOP 0 = 0 . (15b)
Definition 1 (The n 𝑛 n italic_n th homotopy group with base space ΞΌ πœ‡ \mu italic_ΞΌ )

Let ΞΌ πœ‡ \mu italic_ΞΌ be a connected subspace on β„³ β„³ \mathcal{M} caligraphic_M , Ο• 0 subscript italic-Ο• 0 \phi_{0} italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be a point on ΞΌ πœ‡ \mu italic_ΞΌ , and f 𝑓 f italic_f be a mapping from ( S n , s ) superscript 𝑆 𝑛 𝑠 (S^{n},s) ( italic_S start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_s ) to ( β„³ , Ο• 0 ) β„³ subscript italic-Ο• 0 (\mathcal{M},\phi_{0}) ( caligraphic_M , italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) . Then, we define a set of equivalent classes which have base points on ΞΌ πœ‡ \mu italic_ΞΌ as ΞΆ n ( β„³ , ΞΌ ) = { [ f ] ∈ Ο€ n ( β„³ , Ο• ) | βˆ€ Ο• ∈ ΞΌ } fragments subscript 𝜁 𝑛 fragments normal-( M normal-, ΞΌ normal-) fragments normal-{ fragments normal-[ f normal-] subscript πœ‹ 𝑛 fragments normal-( M normal-, Ο• normal-) normal-| superscript italic- for-all Ο• ΞΌ normal-} \zeta_{n}(\mathcal{M},\mu)=\{[f]\in\pi_{n}(\mathcal{M},\phi)|\ \ ^{\forall}% \phi\in\mu\} italic_ΞΆ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) = { [ italic_f ] ∈ italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_Ο• ) | start_POSTSUPERSCRIPT βˆ€ end_POSTSUPERSCRIPT italic_Ο• ∈ italic_ΞΌ } . For any [ f ] ∈ Ο€ n ⁒ ( β„³ , Ο• 0 ) delimited-[] 𝑓 subscript πœ‹ 𝑛 β„³ subscript italic-Ο• 0 [f]\in\pi_{n}(\mathcal{M},\phi_{0}) [ italic_f ] ∈ italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and [ g ] ∈ Ο€ n ⁒ ( β„³ , Ο• 1 ) delimited-[] 𝑔 subscript πœ‹ 𝑛 β„³ subscript italic-Ο• 1 [g]\in\pi_{n}(\mathcal{M},\phi_{1}) [ italic_g ] ∈ italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_Ο• start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , where Ο• 0 , Ο• 1 ∈ ΞΌ subscript italic-Ο• 0 subscript italic-Ο• 1 πœ‡ \phi_{0},\phi_{1}\in\mu italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Ο• start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_ΞΌ , [ f ] delimited-[] 𝑓 [f] [ italic_f ] is equivalent to [ g ] delimited-[] 𝑔 [g] [ italic_g ] if there is a path Ξ· : [ 0 , 1 ] β†’ ΞΌ normal-: πœ‚ normal-β†’ 0 1 πœ‡ \eta:[0,1]\to\mu italic_Ξ· : [ 0 , 1 ] β†’ italic_ΞΌ such that Ξ· ⁒ ( 0 ) = Ο• 0 , Ξ· ⁒ ( 1 ) = Ο• 1 formulae-sequence πœ‚ 0 subscript italic-Ο• 0 πœ‚ 1 subscript italic-Ο• 1 \eta(0)=\phi_{0},\eta(1)=\phi_{1} italic_Ξ· ( 0 ) = italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Ξ· ( 1 ) = italic_Ο• start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , and

[ g ] = [ Ξ· ] - 1 βˆ— [ f ] βˆ— [ Ξ· ] , delimited-[] 𝑔 βˆ— superscript delimited-[] πœ‚ 1 delimited-[] 𝑓 delimited-[] πœ‚ [g]=[\eta]^{-1}\ast[f]\ast[\eta], [ italic_g ] = [ italic_Ξ· ] start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT βˆ— [ italic_f ] βˆ— [ italic_Ξ· ] , (8)

where βˆ— normal-βˆ— \ast βˆ— is defined through Eq. ( 5 ). The equivalent classes of ΞΆ n ⁒ ( β„³ , ΞΌ ) subscript 𝜁 𝑛 β„³ πœ‡ \zeta_{n}(\mathcal{M},\mu) italic_ΞΆ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) under Eq. ( 8 ) form a group which we write as Ο€ n ⁒ ( β„³ , ΞΌ ) subscript πœ‹ 𝑛 β„³ πœ‡ \pi_{n}(\mathcal{M},\mu) italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) . Figure 4 schematically shows the equivalence relation. We call Ο€ n ⁒ ( β„³ , ΞΌ ) subscript πœ‹ 𝑛 β„³ πœ‡ \pi_{n}(\mathcal{M},\mu) italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) a homotopy group with a base space ΞΌ πœ‡ \mu italic_ΞΌ . We write an element of Ο€ n ⁒ ( β„³ , ΞΌ ) subscript πœ‹ 𝑛 β„³ πœ‡ \pi_{n}(\mathcal{M},\mu) italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) by { f } 𝑓 \{f\} { italic_f } instead of [ f ] delimited-[] 𝑓 [f] [ italic_f ] . If ΞΌ = Ο• 0 πœ‡ subscript italic-Ο• 0 \mu=\phi_{0} italic_ΞΌ = italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , Ο€ n ⁒ ( β„³ , ΞΌ ) subscript πœ‹ 𝑛 β„³ πœ‡ \pi_{n}(\mathcal{M},\mu) italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_ΞΌ ) reduces to Ο€ n ⁒ ( β„³ , Ο• 0 ) subscript πœ‹ 𝑛 β„³ subscript italic-Ο• 0 \pi_{n}(\mathcal{M},\phi_{0}) italic_Ο€ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_M , italic_Ο• start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and { f } 𝑓 \{f\} { italic_f } reduces to [ f ] delimited-[] 𝑓 [f] [ italic_f ] .

Definition 1

We say that Οƒ ∈ N ⁒ ( Ξ“ ) = A ⁒ u ⁒ t ⁒ ( G , Ξ“ ) 𝜎 𝑁 normal-Ξ“ 𝐴 𝑒 𝑑 𝐺 normal-Ξ“ \sigma\in N(\Gamma)=Aut(G,\Gamma) italic_Οƒ ∈ italic_N ( roman_Ξ“ ) = italic_A italic_u italic_t ( italic_G , roman_Ξ“ ) is a symmetry of the d 𝑑 d italic_d -periodic framework ( G , Ξ“ , p , Ο€ ) 𝐺 normal-Ξ“ 𝑝 πœ‹ (G,\Gamma,p,\pi) ( italic_G , roman_Ξ“ , italic_p , italic_Ο€ ) when the result of acting by Οƒ 𝜎 \sigma italic_Οƒ on the framework is the same as the result of acting by an isometry s ∈ E ⁒ ( d ) 𝑠 𝐸 𝑑 s\in E(d) italic_s ∈ italic_E ( italic_d ) , that is:

s ∘ p = p ∘ Οƒ 𝑠 𝑝 𝑝 𝜎 s\circ p=p\circ\sigma italic_s ∘ italic_p = italic_p ∘ italic_Οƒ (4)

In other words, we have a commutative diagram

V ⟢ p R d ↓ Οƒ ↓ s V ⟢ p R d 𝑉 superscript ⟢ 𝑝 superscript 𝑅 𝑑 missing-subexpression ↓ absent 𝜎 missing-subexpression ↓ absent 𝑠 missing-subexpression 𝑉 superscript ⟢ 𝑝 superscript 𝑅 𝑑 missing-subexpression \begin{array}[]{lcll}V&\stackrel{{\scriptstyle p}}{{\longrightarrow}}&R^{d}\\ \downarrow\sigma&&\downarrow s\\ V&\stackrel{{\scriptstyle p}}{{\longrightarrow}}&R^{d}\end{array} start_ARRAY start_ROW start_CELL italic_V end_CELL start_CELL start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG italic_p end_ARG end_RELOP end_CELL start_CELL italic_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL ↓ italic_Οƒ end_CELL start_CELL end_CELL start_CELL ↓ italic_s end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_V end_CELL start_CELL start_RELOP SUPERSCRIPTOP start_ARG ⟢ end_ARG start_ARG italic_p end_ARG end_RELOP end_CELL start_CELL italic_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY (5)
DΓ©finition 16

Une algΓ¨bre dupliciale est un triplet ( A , βˆ— , β†– ) 𝐴 normal-βˆ— normal-β†– (A,\ast,\nwarrow) ( italic_A , βˆ— , β†– ) , oΓΉ A 𝐴 A italic_A est un espace vectoriel et βˆ— , β†– : A βŠ— A ⟢ A fragments normal-βˆ— normal-, normal-β†– normal-: A tensor-product A normal-⟢ A \ast,\nwarrow:A\otimes A\longrightarrow A βˆ— , β†– : italic_A βŠ— italic_A ⟢ italic_A , avec les axiomes suivants: pour tout x , y , z ∈ A π‘₯ 𝑦 𝑧 𝐴 x,y,z\in A italic_x , italic_y , italic_z ∈ italic_A ,

{ ( x βˆ— y ) βˆ— z = x βˆ— ( y βˆ— z ) , ( x β†– y ) β†– z = x β†– ( y β†– z ) , ( x βˆ— y ) β†– z = x βˆ— ( y β†– z ) . cases βˆ— βˆ— π‘₯ 𝑦 𝑧 βˆ— π‘₯ βˆ— 𝑦 𝑧 fragments fragments ( x β†– y ) β†– z fragments x β†– fragments ( y β†– z ) , β†– βˆ— π‘₯ 𝑦 𝑧 fragments x βˆ— fragments ( y β†– z ) . \left\{\begin{array}[]{rcl}(x\ast y)\ast z&=&x\ast(y\ast z),\\ (x\nwarrow y)\nwarrow z&=&x\nwarrow(y\nwarrow z),\\ (x\ast y)\nwarrow z&=&x\ast(y\nwarrow z).\end{array}\right. { start_ARRAY start_ROW start_CELL ( italic_x βˆ— italic_y ) βˆ— italic_z end_CELL start_CELL = end_CELL start_CELL italic_x βˆ— ( italic_y βˆ— italic_z ) , end_CELL end_ROW start_ROW start_CELL ( italic_x β†– italic_y ) β†– italic_z end_CELL start_CELL = end_CELL start_CELL italic_x β†– ( italic_y β†– italic_z ) , end_CELL end_ROW start_ROW start_CELL ( italic_x βˆ— italic_y ) β†– italic_z end_CELL start_CELL = end_CELL start_CELL italic_x βˆ— ( italic_y β†– italic_z ) . end_CELL end_ROW end_ARRAY (5)

Une algΓ¨bre dupliciale unitaire A 𝐴 A italic_A est un espace vectoriel A = 𝕂 ⁒ 1 βŠ• A Β― 𝐴 direct-sum 𝕂 1 normal-Β― 𝐴 A=\mathbb{K}1\oplus\overline{A} italic_A = blackboard_K 1 βŠ• Β― start_ARG italic_A end_ARG tel que A Β― normal-Β― 𝐴 \overline{A} Β― start_ARG italic_A end_ARG est une algΓ¨bre dupliciale et oΓΉ on a Γ©tendu les deux opΓ©rations βˆ— normal-βˆ— \ast βˆ— et β†– normal-β†– \nwarrow β†– comme suit: Pour tout a ∈ A π‘Ž 𝐴 a\in A italic_a ∈ italic_A ,

1 βˆ— a = a βˆ— 1 = a , 1 β†– a = a β†– 1 = a . βˆ— 1 π‘Ž βˆ— π‘Ž 1 π‘Ž β†– 1 π‘Ž π‘Ž β†– 1 π‘Ž \begin{array}[]{rcl}1\ast a=a\ast 1&=&a,\\ 1\nwarrow a=a\nwarrow 1&=&a.\end{array} start_ARRAY start_ROW start_CELL 1 βˆ— italic_a = italic_a βˆ— 1 end_CELL start_CELL = end_CELL start_CELL italic_a , end_CELL end_ROW start_ROW start_CELL 1 β†– italic_a = italic_a β†– 1 end_CELL start_CELL = end_CELL start_CELL italic_a . end_CELL end_ROW end_ARRAY
Definition 3 (Decomposable aggregation function) :

An aggregation function f : β„• I β†’ O : 𝑓 β†’ superscript β„• 𝐼 𝑂 f:\mathbb{N}^{I}\to O italic_f : blackboard_N start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT β†’ italic_O is said to be decomposable if for some function g 𝑔 g italic_g and a self-decomposable aggregation function h β„Ž h italic_h , it can be expressed as:

f = g ∘ h 𝑓 𝑔 β„Ž f=g\circ h italic_f = italic_g ∘ italic_h