Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Chains Homotopy in the Complement of a Knot in the Sphere $$S^3$$ S 3

Chains Homotopy in the Complement of a Knot in the Sphere $$S^3$$ S 3 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Chains Homotopy in the Complement of a Knot in the Sphere $$S^3$$ S 3

Loading next page...
 
/lp/springer-journals/chains-homotopy-in-the-complement-of-a-knot-in-the-sphere-s-3-s-3-7s0R9MntIb

References (8)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-019-00136-1
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Feb 21, 2019

There are no references for this article.