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On framings of links in 3-manifolds

Part of: Low-dimensional topology

Published online by Cambridge University Press:  21 September 2020

Rhea Palak Bakshi ,
Dionne Ibarra ,
Gabriel Montoya-Vega ,
Józef H. Przytycki  and
Deborah Weeks
Rhea Palak Bakshi
Affiliation:
Department of Mathematics, The George Washington University, Washington DC, USA e-mail: rhea_palak@gwu.edu dfkunkel@gwu.edu gmontoyavega@gwu.edu deweeks@gwu.edu
Dionne Ibarra
Affiliation:
Department of Mathematics, The George Washington University, Washington DC, USA e-mail: rhea_palak@gwu.edu dfkunkel@gwu.edu gmontoyavega@gwu.edu deweeks@gwu.edu
Gabriel Montoya-Vega
Affiliation:
Department of Mathematics, The George Washington University, Washington DC, USA e-mail: rhea_palak@gwu.edu dfkunkel@gwu.edu gmontoyavega@gwu.edu deweeks@gwu.edu
Józef H. Przytycki*
Affiliation:
Department of Mathematics, The George Washington University, Washington DC, USA and Department of Mathematics, University of Gdańsk, Gdańsk, Poland
Deborah Weeks
Affiliation:
Department of Mathematics, The George Washington University, Washington DC, USA e-mail: rhea_palak@gwu.edu dfkunkel@gwu.edu gmontoyavega@gwu.edu deweeks@gwu.edu
*
e-mail: przytyck@gwu.edu
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Abstract

We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^{2}$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules.

Keywords

Knotslinks3-manifoldsframings of linksskein modulesspin structuresDehn homeomorphismsincompressible surfaces

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 752 - 764
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The fourth author was partially supported by Simons Collaboration Grant-637794 and the CCAS Dean’s Research Chair award.

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