ScalingStacks

To describe the second map ΨI×Y;Δ−,α′\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}, consider the diagram

(12) ⨁α∈⟨α′⟩​𝒮0N​(W1,K⁡(k−,k+)∪L∪J,α){\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\mathcal{S}_{0}^{N}(W_{1};K(k^{-},k^{+})\cup L\cup J,\alpha)}⨁α∈⟨α′⟩​𝒮0N​(W1,K⁡(k−+s−,k++s+)∪L,α){\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\mathcal{S}_{0}^{N}(W_{1};K(k^{-}+s^{-},k^{+}+s^{+})\cup L,\alpha)}⨁α∈⟨α′⟩​𝒮¯0N,α​(W1,K,L∪J){\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L\cup J)}⨁α∈⟨α′⟩​𝒮¯0N,α​(W1,K,L){\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)}⨁α∈⟨α′⟩​𝒮0N​(W2,L∪J,α){\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\mathcal{S}_{0}^{N}(W_{2};L\cup J,\alpha)}⨁α∈⟨α′⟩​𝒮0N​(W2,L,α).{\lx@inpgf@ignorespaces\underset{\alpha\in\langle\alpha^{\prime}\rangle}{\bigoplus}\mathcal{S}_{0}^{N}(W_{2};L,\alpha).}ΨI×∂W1;Σ−,α′\scriptstyle{\lx@inpgf@ignorespaces\Psi_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}}Ψ¯I×∂W1;Σ−,α′\scriptstyle{\lx@inpgf@ignorespaces\underline{\Psi}_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}}Φ\scriptstyle{\lx@inpgf@ignorespaces\Phi}≅\scriptstyle{\lx@inpgf@ignorespaces\cong}Φ\scriptstyle{\lx@inpgf@ignorespaces\Phi}≅\scriptstyle{\lx@inpgf@ignorespaces\cong}ΨI×Y;Δ−,α′\scriptstyle{\lx@inpgf@ignorespaces\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}}

Here, in the top row we wrote (k−,k+)(k^{-},k^{+}) for a pair (r−α−,r−α+)(r-\alpha^{-},r-\alpha^{+}) as in Definition 3.1. The vertical maps from the first to the second row are induced by the inclusion of the summands into the cabled skein lasagna module; cf. Definition 3.1. The vertical maps from the second to the third row are the isomorphisms Φ\Phi from Theorem 3.2.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Ciprian Manolescu, Kevin Walker, Paul Wedrich

Original source: arXiv:2206.04616v2