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If $$\gamma $$ γ is a knot in $$ S^3 \cong \mathbb {H}^2 _{\mathbb {C}} \subset \mathbb {P}_{\mathbb {C}}^2$$ S 3 ≅ H C 2 ⊂ P C 2 , then the set $$\Lambda (\gamma )\subset \mathbb {P}_{\mathbb {C}}^2$$ Λ ( γ ) ⊂ P C 2 is defined as the union of all the complex lines tangent to $$\partial \mathbb {H}^2 _{\mathbb {C}}$$ ∂ H C 2 at points in the image of $$\gamma $$ γ . The following result is obtained: the number of components of $$\Omega (\gamma )=\mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )$$ Ω ( γ ) = P C 2 \ Λ ( γ ) is greater or equal to the number of distinct integers in the set $$\{\ell (\gamma , C): C \text { is a positively oriented chain disjoint to } \gamma \}$$ { ℓ ( γ , C ) : C is a positively oriented chain disjoint to γ } , where $$\ell (\gamma , C)$$ ℓ ( γ , C ) denotes the linking number between $$\gamma $$ γ and C.
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Feb 21, 2019
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