Research ArticleNo Access

Concordance maps in HFK $^{-}$ $^{-}$

Lev Tovstopyat-Nelip

Lev Tovstopyat-Nelip

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

E-mail Address: tovstopy@msu.edu

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https://doi.org/10.1142/S0218216522500316Cited by:0 (Source: Crossref)

Abstract

We show that a decorated knot concordance $𝒞$ from $K_{0}$ to $K_{1}$ induces an $𝔽 [U]$ -module homomorphism

G 𝒞 : HFK - (- S 3, K 0) \to HFK - (- S 3, K 1), <math display="block" altimg="eq-00007.gif"><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="cal">𝒞</mi></mrow></msub><mo class="MathClass-punc">:</mo><msup><mrow><mstyle><mtext class="textrm" mathvariant="normal">HFK</mtext></mstyle></mrow><mrow><mo stretchy="false" class="MathClass-bin">-</mo></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">-</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo class="MathClass-punc">,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\to</mo><msup><mrow><mstyle><mtext class="textrm" mathvariant="normal">HFK</mtext></mstyle></mrow><mrow><mo stretchy="false" class="MathClass-bin">-</mo></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">-</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo class="MathClass-punc">,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mrow></math>

which preserves the Alexander and absolute

$ℤ_{2}$ -Maslov gradings. Our construction generalizes the concordance maps induced on

$\hat{HFK}$ studied by Juhász and Marengon [Concordance maps in knot Floer homology, Geom. Topol.20 (2016) 3623–3673], but uses the description of

${HFK}^{-}$ as a direct limit of maps between sutured Floer homology groups discovered by Etnyre et al. [Sutured Floer homology and invariants of Legendrian and transverse knots, Geom. Topol.21 (2017) 1469–1582].

Keywords:

AMSC: 57M27, 57R58

References

1. A. Alishahi and E. Eftekhary , Tangle Floer homology and cobordisms between tangles, J. Topol. 13(4) (2020) 1582–1657. Crossref, Web of Science, Google Scholar
2. G. Arone and M. Kankaanrinta , On the functoriality of the blow-up construction, Bull. Belg. Math. Soc. Simon Stevin 17(5) (y2010) 821–832. Crossref, Web of Science, Google Scholar
3. M. Atiyah , Topological quantum field theories, Publ. Math. Inst. Hautes Études Sci. 68 (1988) 175–186. Crossref, Google Scholar
4. J. Baldwin and S. Sivek , Invariants of Legendrian and transverse knots in monopole knot homology, J. Symplectic Geom. 16 (2014) 959–1000. Crossref, Web of Science, Google Scholar
5. C. Blanchet and V. Turaev , Axiomatic approach to TQFT, in Encyclopedia of Mathematical Physics (Elsevier, 2006), pp. 232–234. Crossref, Google Scholar
6. J. B. Etnyre, D. S. Vela-Vick and R. Zarev , Sutured Floer homologyand invariants of Legendrian and transverse knots, Geom. Topol. 21 (2017) 1469–1582. Crossref, Web of Science, Google Scholar
7. D. Gabai , Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983) 445–503. Crossref, Web of Science, Google Scholar
8. K. Honda , On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309–368. Crossref, Web of Science, Google Scholar
9. K. Honda, W. Kazez and G. Matic̀, Contact structures, sutured Floer homology, and TQFT, preprint (2008), arXiv:math/0807.2431. Google Scholar
10. A. Juhász , Holomorphic disks and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429–1457. Crossref, Web of Science, Google Scholar
11. A. Juhász , Cobordisms of sutured manifolds and the functoriality of link Floer homology, Adv. Math. 299 (2016) 940–1038. Crossref, Web of Science, Google Scholar
12. A. Juhász and M. Marengon , Concordance maps in knot Floer homology, Geom. Topol. 20 (2016) 3623–3673. Crossref, Web of Science, Google Scholar
13. A. Juhász and M. Marengon , Computing cobordism maps in link Floer homologyand the reduced Khovanov TQFT, Selecta Math. 24 (2018) 1315–1390. Crossref, Web of Science, Google Scholar
14. R. Lutz , Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier 27 (1977) 11–15. Crossref, Google Scholar
15. P. Ozsváth and Z. Szabó , Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116. Crossref, Web of Science, Google Scholar
16. P. Ozsváth and Z. Szabó , Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158. Crossref, Web of Science, Google Scholar
17. P. S. Ozsváth, Z. Szabó, P. Lisca and A. I. Stipsicz , Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. 11 (2009) 1307–1363. Crossref, Web of Science, Google Scholar
18. J. A. Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University, MA, USA (2003). Google Scholar
19. I. Zemke , Link cobordisms and functoriality in link Floer homology, J. Topol. 12 (2018) 94–220. Crossref, Web of Science, Google Scholar