Moving basepoints and the induced automorphisms of link Floer homology

Sarkar, Sucharit

doi:10.2140/agt.2015.15.2479

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Moving basepoints and the induced automorphisms of link Floer homology

Sucharit Sarkar

Algebraic & Geometric Topology 15 (2015) 2479–2515

DOI: 10.2140/agt.2015.15.2479

Abstract

Given an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>l</mi><mspace width="0.3em" class="thinspace"/></math>$ –component pointed oriented link $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><mi>L</mi><mo class="MathClass-punc">,</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ in an oriented three-manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Y</mi> </math>$ , one can construct its link Floer chain complex $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="italic"><mi>C</mi><mi>F</mi><mi>L</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi><mo class="MathClass-punc">,</mo><mi>L</mi><mo class="MathClass-punc">,</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ over the polynomial ring $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo class="MathClass-open">[</mo><mrow><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo class="MathClass-op">…</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow><mo class="MathClass-close">]</mo></mrow></math>$ . Moving the basepoint $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub> <mo class="MathClass-rel">∈</mo> <msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math>$ once around the link component $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math>$ induces an automorphism of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="italic"><mi>C</mi><mi>F</mi><mi>L</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi><mo class="MathClass-punc">,</mo><mi>L</mi><mo class="MathClass-punc">,</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ . We study a (possibly different) automorphism of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="italic"><mi>C</mi><mi>F</mi><mi>L</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi><mo class="MathClass-punc">,</mo><mi>L</mi><mo class="MathClass-punc">,</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ defined explicitly in terms of holomorphic disks; for links in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math>$ , we show that these two automorphisms are the same.

Keywords

link Floer homology, basepoint, mapping class group action, grid diagram

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27, 57R58

References

Publication

Received: 18 September 2011

Revised: 25 June 2014

Accepted: 2 March 2015

Published: 10 December 2015

Authors

Sucharit Sarkar
Department of Mathematics Princeton University Fine Hall Washington Rd Princeton, NJ 08544 USA
http://web.math.princeton.edu/~sucharit/

Volume 15, issue 5 (2015)