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Chains Homotopy in the Complement of a Knot in the Sphere \(S^3\)

  • W. BarreraEmail author
  • A. Cano
  • R. García
  • J. P. Navarrete
Article
  • 75 Downloads

Abstract

If \(\gamma \) is a knot in \( S^3 \cong \mathbb {H}^2 _{\mathbb {C}} \subset \mathbb {P}_{\mathbb {C}}^2\), then the set \(\Lambda (\gamma )\subset \mathbb {P}_{\mathbb {C}}^2\) is defined as the union of all the complex lines tangent to \(\partial \mathbb {H}^2 _{\mathbb {C}}\) at points in the image of \(\gamma \). The following result is obtained: the number of components of \(\Omega (\gamma )=\mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )\) is greater or equal to the number of distinct integers in the set \(\{\ell (\gamma , C): C \text { is a positively oriented chain disjoint to } \gamma \}\), where \(\ell (\gamma , C)\) denotes the linking number between \(\gamma \) and C.

Keywords

Complex hyperbolic plane Complex projective plane Chain Knot 

Mathematics Subject Classification

51M10 57M25 

Notes

Acknowledgements

We would like to thank Professors John Parker and José Seade for enlightening discussions during the process of this work and the kind hospitality received by the three authors at IMATE-UNAM -Cuernavaca, and the Facultad de Matemáticas, UADY.

References

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Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad Autónoma de Yucatán, Anillo Periférico Norte Tablaje CatMéridaMexico
  2. 2.Instituto de Matemáticas UNAMCuernavacaMexico
  3. 3.School of MathematicsUniversity of LeedsLeedsUK

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