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Nonfillable Legendrian knots in the 3-sphere. (English) Zbl 1387.57041
If a Legendrian knot \(\Lambda\) in the standard contact 3-sphere bounds an exact orientable Lagrangian surface \(\Sigma\) in the standard symplectic 4-ball, then the genus of \(\Lambda\) is equal to the slice genus of the smooth knot underlying \(\Lambda\). In [Algebr. Geom. Topol. 10, No. 1, 63–85 (2010; Zbl 1203.57010)], B. Chantraine showed that the rotation number of \(\Lambda\) is zero as well as the sum of the Thurston-Bennequin numbers of \(\Lambda\) and the Euler characteristic of \(\Sigma\). In [Prog. Math. 296, 109–145 (2012; Zbl 1254.57024)], T. Ekholm showed that the linearized contact homology of \(\Lambda\) with respect to the augmentation induced by \(\Sigma\) is isomorphic to the singular homology of \(\Lambda\). In [J. Eur. Math. Soc. (JEMS) 18, No. 11, 2627–2689 (2016; Zbl 1357.57044)], T. Ekholm, K. Honda, and T. Kálmán asked if every augmentation for which Seidel’s isomorphism holds is induced by a Lagrangian filling.
In this paper, the author gives a negative answer to this question by considering the family of Legendrian knots \(\{\Lambda_{p,q,r,s}\}\) with \(p\equiv q\equiv r\equiv s(\text{mod } 2)\). The rotation and Thurston-Bennequin numbers of \(\Lambda_{p,q,r,s}\) are \(0\) and \(5\), respectively. This gives a lower bound of \(3\) on the slice genus. On the other hand, the Seifert surface the author obtained from this projection has genus \(3\). Hence, both the slice genus and the Seifert genus are \(3\). The author proves that the Chekanov-Eliashberg dg-algebra of \(\Lambda_{p,p,p+1,p+1}\) admits an augmentation which is not induced by any exact orientable Lagrangian filling, although the corresponding linearized contact homology is isomorphic to the homology of a surface of genus \(3\) with one boundary component.
MSC:
57R17 Symplectic and contact topology in high or arbitrary dimension
16E45 Differential graded algebras and applications (associative algebraic aspects)
53D42 Symplectic field theory; contact homology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:
[1] 10.4310/JSG.2014.v12.n3.a5 · Zbl 1308.53119
[2] 10.2140/agt.2010.10.63 · Zbl 1203.57010
[3] 10.1007/s002220200212 · Zbl 1029.57011
[4] 10.1017/S0305004110000460 · Zbl 1243.57009
[5] 10.1007/BF01389853 · Zbl 0312.55011
[6] 10.4171/QT/73 · Zbl 1346.53074
[7] 10.1007/978-0-8176-8277-4_6 · Zbl 1254.57024
[8] 10.4171/JEMS/650 · Zbl 1357.57044
[9] 10.2140/gt.2017.21.3313 · Zbl 1378.57041
[10] 10.4171/063 · Zbl 1058.05063
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