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Non-decomposable Lagrangian cobordisms between Legendrian knots

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  10 August 2026

ROMAN GOLOVKO Open the ORCID record for ROMAN GOLOVKO [Opens in a new window]  and
DANIEL KOMÁREK
ROMAN GOLOVKO
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 49/83, Prague 8, 186 00, Czech Republic. e-mail: golovko@karlin.mff.cuni.cz
DANIEL KOMÁREK
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 49/83, Prague 8, 186 00, Czech Republic. e-mail: daniel.komarek@matfyz.cuni.cz
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Abstract

For a given $g\gt0$, we construct a family of non-decomposable Lagrangian cobordisms of genus g between (stabilised) Legendrian knots in the standard contact three-sphere. The main technique we use to obstruct decomposability is based on Livingston’s estimates.

MSC classification

Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53D42: Symplectic field theory; contact homology

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 7
Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The Cambridge Philosophical Society

1. Introduction and main results

The standard way to construct Lagrangian cobordisms between Legendrian knots in the standard contact upper S Subscript s t d Superscript 3 $S^3_{std}$ is by concatenating elementary cobordisms which are traces of three moves in the front projection: Legendrian isotopy, isolated standard unknot birth (induces Lagrangian 0-handle attachment), and Legendrian surgery, which is called a pinch move in reverse (it induces Lagrangian 1-handle attachment). See Figure 1 for the schematic description of the latter two pieces. Cobordisms obtained this way are called decomposable and have been first constructed by Chantraine in [ Reference Chantraine4 ] and Ekholm–Honda-Kalman in [ Reference Ekholm, Honda and Kálmán10 ].

Fig. 1.

Left: Lagrangian filling of the t b equals negative 1 $tb=-1$ unknot; Right: Lagrangian pair-of-pants cobordism obtained from the Lagrangian 1-handle attachment.

Illustration of Lagrangian cobordisms between Legendrian knots, showing two different configurations.

Decomposable Lagrangian cobordisms can be seen as symplectic analogues of ribbon cobordisms in smooth topology. Recall that for a given two knots upper K 0 comma upper K 1 subset of upper S cubed $K_0,K_1\subset S^3$ a ribbon cobordism normal upper Sigma $\Sigma$ from upper K 0 $K_0$ to upper K 1 $K_1$ is a smooth cobordism normal upper Sigma subset of upper S cubed times left bracket 0 comma 1 right bracket $\Sigma\subset S^3\times[0,1]$ from upper K 0 $K_0$ to upper K 1 $K_1$ such that the projection upper S cubed times left bracket 0 comma 1 right bracket right arrow left bracket 0 comma 1 right bracket $S^3\times [0,1]\to [0,1]$ restricts to a Morse function on normal upper Sigma $\Sigma$ which does not have index two critical points. Note that all decomposable Lagrangian cobordisms between Legendrian knots in upper S Subscript s t d Superscript 3 $S^3_{std}$ are ribbon.

These days, non-decomposable Lagrangian cobordisms are proven to exist in the following two forms:

  1. (1) non-decomposable Lagrangian caps with stabilised Legendrian ends that have been discussed in the work of Lin [ Reference Lin13 ];

  2. (2) non-decomposable Lagrangian concordances between stabilised Legendrian knots [ Reference Dimitroglou Rizell and Golovko8 ] and between Lagrangian fillable Legendrian knots [ Reference Dimitroglou Rizell and Golovko9 ] that have been constructed by Dimitroglou Rizell and the first author.

In this paper we construct non-decomposable Lagrangian cobordisms of an arbitrary non-trivial genus between Legendrian knots in upper S cubed $S^3$ . The method to determine non-decomposability that we use here is different from the ones we used with Dimitroglou Rizell in [ Reference Dimitroglou Rizell and Golovko8, Reference Dimitroglou Rizell and Golovko9 ]. Recall that in [ Reference Dimitroglou Rizell and Golovko8 ] we use the obstruction of non-decomposability based on the variation of the result of Cornwell-Ng-Sivek [ Reference Cornwell, Ng and Sivek6 , theorem 3·2], and in [ Reference Dimitroglou Rizell and Golovko9 ] we relied on the result of Agol which says that ribbon (stably homotopy ribbon) Lagrangian concordances of knots form a partial order, see [ Reference Agol1 , theorem 1·1]. In this paper, we rely on the obstruction of ribbon cobordisms due to Livingston [ Reference Livingston14 , corollary 5·4] and its partial adaptation due to Borodzik and Truöl [ Reference Borodzik and Truöl3 , theorem 2·5].

The main result of this paper is the following:

Theorem 1·1. Let normal upper Lamda $\Lambda$ be a Legendrian knot in left parenthesis upper S cubed comma xi Subscript s t Baseline right parenthesis $(S^3, \xi_{st})$ such that:

  1. (i) there is an odd prime p that divides the determinant of normal upper Lamda $\Lambda$ ;

  2. (ii) there is a decomposable Lagrangian concordance C from the t b equals negative 1 $tb=-1$ unknot U to normal upper Lamda $\Lambda$ .

Then for a given g greater than 0 $g \gt 0$ , and for all integers n greater than 2 g $n \gt 2g$ , there exists a non-decomposable Lagrangian cobordism upper L Subscript g $L_g$ of genus g, from a Legendrian knot normal upper Lamda Subscript minus $\Lambda_{-}$ obtained from normal upper Lamda Superscript n Baseline colon equals normal upper Lamda number sign ellipsis number sign normal upper Lamda $\Lambda^n:=\Lambda\#\dots\#\Lambda$ by sufficiently many stabilisations, to a Legendrian knot normal upper Lamda Subscript plus $\Lambda_+$ obtained from upper U Superscript n Baseline colon equals upper U number sign ellipsis number sign upper U $U^n:=U\#\dots\#U$ by sufficiently many stabilisations.

Remark 1·2. Note that det left parenthesis upper U right parenthesis equals 1 $\det(U)=1$ , and hence condition (i) in the formulation of Theorem 1·1 implies that normal upper Lamda $\Lambda$ is not smoothly isotopic to U.

2. Applicability of Theorem 1·1

Here we describe a few series of examples to which we can apply Theorem 1·1.

Examples 2·1. Consider the sequence of Legendrian pretzel knots upper P left parenthesis 3 comma negative 3 comma negative k right parenthesis $P(3,-3,-k)$ , where k greater than or equals 3 $k\geq 3$ , from [ Reference Chantraine and Legout5, Reference Dimitroglou Rizell and Golovko8 ], we call them normal upper Lamda Subscript k $\Lambda_{k}$ . There is a decomposable Lagrangian concordance upper C Subscript k $C_{k}$ from U to normal upper Lamda Subscript k $\Lambda_{k}$ , see Figure 2, this concordance has been discussed by Chantraine and Legout in [ Reference Chantraine and Legout5 ] and by Dimitroglou Rizell and the first author in [ Reference Dimitroglou Rizell and Golovko8 ]. Following the computation of the determinant for pretzel knots from [ Reference Bae and Lee2 ] we observe that det left parenthesis upper P left parenthesis 3 comma negative 3 comma negative k right parenthesis right parenthesis equals 9 $\det(P(3,-3,-k))=9$ and the odd prime p divisor of det left parenthesis upper P left parenthesis 3 comma negative 3 comma negative k right parenthesis right parenthesis $\det(P(3,-3,-k))$ that is required by Theorem 1·1 can be taken to be p equals 3 $p=3$ . Hence, Theorem 1·1 can be applied to upper C Subscript k $C_k$ .

Fig. 2.

Decomposable Lagrangian concordance upper C Subscript k $C_{k}$ from the t b equals negative 1 $tb=-1$ Legendrian unknot U to the Legendrian representative normal upper Lamda Subscript k $\Lambda_{k}$ of the pretzel knot upper P left parenthesis 3 comma negative 3 comma negative k right parenthesis $P(3,-3,-k)$ , where k greater than or equals 3 $k\geq 3$ , induced by the ambient surgery along the dashed arc. The front projection of normal upper Lamda Subscript k $\Lambda_{k}$ has k minus 3 $k-3$ crossings in the dashed box.

A diagram illustrating a mathematical concept involving knots and projections.

Examples 2·2. In order to describe the following family we first recall a few properties of determinant and Alexander polynomial.

Remark 2·3. Determinant and Alexander polynomial satisfy the following properties:

  1. (1) For a given knot K, StartAbsoluteValue normal upper Delta Subscript upper K Baseline left parenthesis negative 1 right parenthesis EndAbsoluteValue equals det left parenthesis upper K right parenthesis $|\Delta_{K}(-1)|=\det(K)$ ;

  2. (2) For given two knots upper K 1 $K_1$ and upper K 2 $K_2$ , the Alexander polynomial of the connected sum upper K 1 number sign upper K 2 $K_1\# K_2$ is given by normal upper Delta Subscript upper K 1 number sign upper K 2 Baseline left parenthesis t right parenthesis equals normal upper Delta Subscript upper K 1 Baseline left parenthesis t right parenthesis normal upper Delta Subscript upper K 2 Baseline left parenthesis t right parenthesis $\Delta_{K_1\# K_2}(t) = \Delta_{K_1}(t)\Delta_{K_2}(t)$ , which implies that

    det left parenthesis upper K 1 number sign upper K 2 right parenthesis equals StartAbsoluteValue normal upper Delta Subscript upper K 1 number sign upper K 2 Baseline left parenthesis negative 1 right parenthesis EndAbsoluteValue equals StartAbsoluteValue normal upper Delta Subscript upper K 1 Baseline left parenthesis negative 1 right parenthesis EndAbsoluteValue StartAbsoluteValue normal upper Delta Subscript upper K 2 Baseline left parenthesis negative 1 right parenthesis EndAbsoluteValue equals det left parenthesis upper K 1 right parenthesis d e t left parenthesis upper K 2 right parenthesis semicolon $$\det(K_1\# K_2) = |\Delta_{K_1\# K_2}(-1)| = |\Delta_{K_1}(-1)||\Delta_{K_2}(-1)| = \det(K_1)det(K_2);$$

  3. (3) For a knot K, det left parenthesis upper K right parenthesis not equals 0 $\det(K)\neq 0$ . This, in particular, can be justified by the fact that det left parenthesis upper K right parenthesis $\det(K)$ can be seen as the order of the finite abelian group upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper K right parenthesis semicolon double struck upper Z right parenthesis $H_1(\Sigma_2(K);\mathbb Z)$ , where normal upper Sigma 2 left parenthesis upper K right parenthesis $\Sigma_2(K)$ denotes the 2-fold branched cover of upper S cubed $S^3$ over K.

Now we take a Legendrian knot normal upper Lamda $\Lambda$ such that there is a decomposable Lagrangian concordance upper C prime $C'$ from U to normal upper Lamda $\Lambda$ . Then we take the connected sum of decomposable Lagrangian concordances upper C Subscript k Baseline number sign upper C prime $C_{k}\# C'$ from upper U number sign upper U asymptotically equals upper U $U\# U\simeq U$ to normal upper Lamda Subscript k Baseline number sign normal upper Lamda $\Lambda_{k}\# \Lambda$ . Such an operation can be performed following the discussion in [ Reference Guadagni, Sabloff and Yacavone12 , section 5]. Observe that upper C Subscript k Baseline number sign upper C prime $C_{k}\# C'$ is decomposable by [ Reference Guadagni, Sabloff and Yacavone12 , corollary 5·1, remark 5·3]. Then note that from the third part of Remark 2·3 it follows that det left parenthesis normal upper Lamda right parenthesis not equals 0 $\det(\Lambda)\neq 0$ . From this, together with the second part of Remark 2·3, it follows that p equals 3 vertical bar det left parenthesis normal upper Lamda Subscript k Baseline number sign normal upper Lamda right parenthesis $p=3 |\det(\Lambda_{k}\# \Lambda)$ . So, we can apply Theorem 1·1 to the concordance upper C Subscript k Baseline number sign upper C prime $C_{k}\# C'$ .

Remark 2·4. For example, we can take upper C prime $C'$ to be a decomposable Lagrangian concordance from U to the Legendrian representative of (2,n)-torus knot from [ Reference Pan15 ].

Examples 2·5. Here we rely on the decomposable realisation of ribbon cobordisms developed by Etnyre and Leverson in [ Reference Etnyre and Leverson11 ]. We can simply take a smooth knot K satisfying that p vertical bar d e t left parenthesis upper K right parenthesis $p | det(K)$ for some odd prime p $p$ and such that there is a ribbon concordance from an unknot to K. Then following [ Reference Etnyre and Leverson11 , theorem 1·2] we realize this concordance by a decomposable Lagrangian concordance C from the Legendrian representative of an unknot (in general, stabilised) to the Legendrian representative of K (in general, stabilised). Obviously, Theorem 1·1 can be applied to C.

3. Proof of Theorem 1·1

We start the proof by considering the connected sum of n copies of concordance C and denote it by upper C Superscript n Baseline colon equals upper C number sign ellipsis number sign upper C $C^n:=C\#\ \dots \# C$ . Since C is a decomposable Lagrangian concordance, from [ Reference Guadagni, Sabloff and Yacavone12 , corollary 5·1, remark 5·3] it follows that upper C Superscript n $C^n$ is a decomposable Lagrangian concordance from the Legendrian knot upper U Superscript n Baseline equals upper U number sign ellipsis number sign upper U $U^n=U\#\dots \#U$ , which is smoothly an unknot, to the Legendrian knot normal upper Lamda Superscript n Baseline equals normal upper Lamda number sign ellipsis number sign normal upper Lamda $\Lambda^n=\Lambda\# \dots \# \Lambda$ . We know that there is a prime number p that divides det left parenthesis normal upper Lamda right parenthesis $\det(\Lambda)$ . From the second part of Remark 2·3 we observe that det left parenthesis normal upper Lamda Superscript n Baseline right parenthesis equals det left parenthesis normal upper Lamda right parenthesis Superscript n $\det(\Lambda^n)=\det(\Lambda)^n$ , and hence p that divides det left parenthesis normal upper Lamda right parenthesis $\det(\Lambda)$ also divides det left parenthesis normal upper Lamda Superscript n Baseline right parenthesis $\det(\Lambda^n)$ and does not divide det left parenthesis upper U Superscript n Baseline right parenthesis equals det left parenthesis upper U right parenthesis equals 1 $\det(U^n)=\det(U)=1$ .

Then we argue the same way as in [ Reference Dimitroglou Rizell and Golovko9 , section 2] and [ Reference Dimitroglou Rizell and Golovko8 , section 5]. First, we revert the concordance left parenthesis upper C Superscript n Baseline right parenthesis Superscript negative 1 Baseline equals StartSet left parenthesis negative t comma x right parenthesis element of double struck upper R Subscript t Baseline times double struck upper R Subscript x Superscript 3 Baseline vertical bar left parenthesis t comma x right parenthesis element of upper C Superscript n Baseline EndSet period $(C^n)^{-1}=\{(-t,x)\in \mathbb{R}_t\times \mathbb{R}^3_x\ | \ (t,x)\in C^n\}.$ Since upper C Superscript n $C^n$ is decomposable, observe that the tangent planes of left parenthesis upper C Superscript n Baseline right parenthesis Superscript negative 1 $(C^n)^{-1}$ are homotopic to totally-real tangent planes. Using this, we apply the h-principle for totally real embeddings and see that left parenthesis upper C Superscript n Baseline right parenthesis Superscript negative 1 $(C^n)^{-1}$ admits a compactly supported smooth isotopy to a concordance that is totally real for any choice of cylindrical almost complex structure, we denote it by upper T Superscript n $T^n$ . Finally, we modify upper T Superscript n $T^n$ using the approximation result of Dimitroglou Rizell [ Reference Dimitroglou Rizell7 ] and get the Lagrangian concordance upper S left parenthesis upper T Superscript n Baseline right parenthesis $S(T^n)$ from upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis $S(\Lambda^n)$ to upper S left parenthesis upper U Superscript n Baseline right parenthesis $S(U^n)$ . Here upper S left parenthesis upper U Superscript n Baseline right parenthesis $S(U^n)$ and upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis $S(\Lambda^n)$ are multiply stabilised versions of upper U Superscript n $U^n$ and normal upper Lamda Superscript n $\Lambda^n$ , respectively.

3·1. Lagrangian cobordisms constructed by Sabloff, Vela-Vick and Wong

Sabloff, Vela-Vick and Wong in [ Reference Sabloff, Vela-Vick and Wong16 ] constructed the following cobordism that will be useful for us. Given a Legendrian knot normal upper Lamda prime subset of upper S Subscript s t d Superscript 3 $\Lambda'\subset S^3_{std}$ , let left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis left parenthesis normal upper Lamda Superscript prime Baseline right parenthesis $(S_+S_-)(\Lambda')$ be the Legendrian knot obtained by first applying the negative stabilisation to normal upper Lamda prime $\Lambda'$ followed by the application of the positive stabilisation. Sabloff, Vela-Vick and Wong in [ Reference Sabloff, Vela-Vick and Wong16 , lemma 3·2] showed that there is a Lagrangian cobordism of genus one from left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis left parenthesis normal upper Lamda Superscript prime Baseline right parenthesis $(S_+S_-)(\Lambda')$ to normal upper Lamda prime $\Lambda'$ , the schematic picture of it appears in Figure 3.

Fig. 3.

Lagrangian cobordism of genus one from left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis left parenthesis normal upper Lamda Superscript prime Baseline right parenthesis $(S_+S_-)(\Lambda')$ to normal upper Lamda prime $\Lambda'$ obtained by making an ambient surgery along two dashed arcs.

A diagram illustrating Lagrangian cobordism of genus one through ambient surgery.

We apply this construction inductively to normal upper Lamda prime equals upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis $\Lambda'=S(\Lambda^n)$ . In other words, given g greater than 0 $g \gt 0$ , we concatenate g copies of the genus-1 Lagrangian cobordism from [ Reference Sabloff, Vela-Vick and Wong16 , lemma 3·2] to obtain a genus-g Lagrangian cobordism from left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis $(S_+S_-)^g(S(\Lambda^n))$ to upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis $S(\Lambda^n)$ . Note that the genus is additive because the cobordisms that we are concatenating have connected boundaries.

We then concatenate the obtained cobordism and upper S left parenthesis upper T Superscript n Baseline right parenthesis $S(T^n)$ and get a Lagrangian cobordism upper L Subscript g $L_g$ of genus g from normal upper Lamda Subscript minus Baseline equals left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis $\Lambda_-=(S_+S_-)^g(S(\Lambda^n))$ to normal upper Lamda Subscript plus Baseline equals upper S left parenthesis upper U Superscript n Baseline right parenthesis $\Lambda_+=S(U^n)$ . We claim that upper L Subscript g $L_g$ is a non-ribbon cobordism. In order to show it, we need to rely on the estimates of Livingston from [ Reference Livingston14 ].

3·2. Livingston’s estimates and the results of Borodzik–Truöl

For a given two knots upper K 0 $K_0$ and upper K 1 $K_1$ in upper S cubed $S^3$ and a connected Morse cobordism normal upper Sigma subset of upper S cubed times left bracket 0 comma 1 right bracket $\Sigma\subset S^3\times [0,1]$ from upper K 0 $K_0$ to upper K 1 $K_1$ (i.e. normal upper Sigma $\Sigma$ is a connected cobordism satisfying that the projection of normal upper Sigma $\Sigma$ to [0,1] is a Morse function), Livingston in [ Reference Livingston14 ] investigates the behaviour of quadruples left parenthesis c 0 comma c 1 comma c 2 comma g right parenthesis $(c_0,c_1,c_2,g)$ , where c Subscript i Baseline equals c Subscript i Baseline left parenthesis normal upper Sigma right parenthesis $c_i=c_i(\Sigma)$ denotes the number of critical points of index i on normal upper Sigma $\Sigma$ and g is a genus of normal upper Sigma $\Sigma$ . More precisely, Livingston in [ Reference Livingston14 ] proved a few obstructions to the existence of normal upper Sigma $\Sigma$ in the form of estimates involving the homology of the n-fold branched covers normal upper Sigma Subscript n Baseline left parenthesis upper K Subscript i Baseline right parenthesis $\Sigma_n(K_i)$ of upper S cubed $S^3$ over upper K Subscript i $K_i$ , i equals 0 comma 1 $i=0,1$ .

In this note we rely on the partial case of Livingston’s estimates that uses the homology of the two-fold branched covers normal upper Sigma 2 left parenthesis upper K 0 right parenthesis $\Sigma_2(K_0)$ and normal upper Sigma 2 left parenthesis upper K 1 right parenthesis $\Sigma_2(K_1)$ .

Theorem 3·1. ([ Reference Livingston14 ]). If normal upper Sigma $\Sigma$ is a connected Morse cobordism from upper K 0 $K_0$ to upper K 1 $K_1$ , then for an arbitrary odd prime p the following inequality holds

(3·1) StartLayout 1st Row c 2 left parenthesis normal upper Sigma right parenthesis greater than or equals StartFraction beta 1 left parenthesis normal upper Sigma 2 left parenthesis upper K 0 right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis minus beta 1 left parenthesis normal upper Sigma 2 left parenthesis upper K 1 right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis Over 2 EndFraction minus g left parenthesis normal upper Sigma right parenthesis period EndLayout \begin{align}c_2(\Sigma)\geq \frac{\beta_1(\Sigma_2(K_0);\; \mathbb F_p)-\beta_1(\Sigma_2(K_1);\; \mathbb F_p)}{2}-g(\Sigma).\end{align}

Here beta 1 left parenthesis normal upper Sigma 2 left parenthesis upper K Subscript i Baseline right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis $\beta_1(\Sigma_2(K_i);\; \mathbb F_p)$ is the dimension of upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper K Subscript i Baseline right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis $H_1(\Sigma_2(K_i);\; \mathbb F_p)$ as an double struck upper F Subscript p $\mathbb F_p$ -vector space, i equals 0 comma 1 $i=0,1$ .

Note that our handle convention is opposite to that of Livingston in [ Reference Livingston14 ]: Livingston considers cobordisms from upper K 1 $K_1$ to upper K 0 $K_0$ , whereas we consider cobordisms from upper K 0 $K_0$ to upper K 1 $K_1$ , and the handle indices are reversed.

Besides that, we need the following statement. It is a standard exercise, and we refer to the work of Borodzik–Truöl [ Reference Borodzik and Truöl3 ] for a proof.

Proposition 3·2 ([ Reference Borodzik and Truöl3 ]). Given a knot upper K subset of upper S cubed $K\subset S^3$ and a prime p.

  1. (i) If p divides det left parenthesis upper K right parenthesis $\det(K)$ , then dimension Subscript double struck upper F Sub Subscript p Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper K Superscript n Baseline equals upper K number sign ellipsis number sign upper K right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis greater than or equals n $\dim_{\mathbb F_p}H_1(\Sigma_2(K^n=K\#\dots \# K);\; \mathbb F_p)\geq n$ .

  2. (ii) If p does not divide det left parenthesis upper K right parenthesis $\det(K)$ , then upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper K right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis equals 0 $H_1(\Sigma_2(K);\; \mathbb F_p)=0$ .

3·3. Application of Livingston’s estimates to obstruct decomposability

Now we take the Lagrangian cobordism upper L Subscript g $L_g$ of genus g from left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis $(S_+S_-)^g(S(\Lambda^n))$ to upper S left parenthesis upper U Superscript n Baseline right parenthesis $S(U^n)$ that we constructed in Section 3·1. From the second part of Remark 2·3 we observe that

det left parenthesis left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis right parenthesis equals det left parenthesis normal upper Lamda right parenthesis Superscript n Baseline and det left parenthesis upper S left parenthesis upper U Superscript n Baseline right parenthesis right parenthesis equals 1 period $$\det((S_+S_-)^g(S(\Lambda^n)))=\det(\Lambda)^n\quad \mbox{and}\quad \det(S(U^n))=1.$$

Besides that, left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis $(S_+S_-)^g(S(\Lambda^n))$ is smoothly isotopic to normal upper Lamda Superscript n $\Lambda^n$ . Hence, using the fact that p vertical bar det left parenthesis normal upper Lamda right parenthesis $p| \det(\Lambda)$ , from the first part of Proposition 3·2 it follows that

(3·2) StartLayout 1st Row dimension Subscript double struck upper F Sub Subscript p Subscript Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis left parenthesis upper S Subscript plus Baseline upper S Subscript minus Baseline right parenthesis Superscript g Baseline left parenthesis upper S left parenthesis normal upper Lamda Superscript n Baseline right parenthesis right parenthesis right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis equals dimension Subscript double struck upper F Sub Subscript p Subscript Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis normal upper Lamda Superscript n Baseline right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis greater than or equals n period EndLayout \begin{align}\dim_{\mathbb F_p}H_1(\Sigma_2((S_+S_-)^g(S(\Lambda^n)));\; \mathbb F_p)=\dim_{\mathbb F_p}H_1(\Sigma_2(\Lambda^n);\; \mathbb F_p)\geq n.\end{align}

Besides that, the second part of Proposition 3·2 implies that

(3·3) StartLayout 1st Row upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper S left parenthesis upper U Superscript n Baseline right parenthesis right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis equals upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper U right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis equals 0 comma and hence dimension Subscript double struck upper F Sub Subscript p Subscript Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis upper U right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis equals 0 period EndLayout \begin{align}H_1(\Sigma_2(S(U^n));\; \mathbb F_p)=H_1(\Sigma_2(U);\; \mathbb F_p)=0,\quad \mbox{and hence}\quad \dim_{\mathbb F_p} H_1(\Sigma_2(U);\; \mathbb F_p)=0.\end{align}

Finally, combining Formulae 3·1, 3·2 and 3·3, we get that c 2 left parenthesis upper L Subscript g Baseline right parenthesis greater than or equals n divided by 2 minus g $c_2(L_g)\geq {n}/{2}-g$ . Therefore, for any fixed g greater than 0 $g\gt0$ , since n greater than 2 g $n\gt2g$ , we see that c 2 left parenthesis upper L Subscript g Baseline right parenthesis greater than 0 $c_2(L_g)\gt0$ . So, we see that upper L Subscript g $L_g$ is a non-ribbon Lagrangian cobordism of genus g greater than 0 $g\gt0$ , and hence upper L Subscript g $L_g$ is a non-decomposable Lagrangian cobordism of genus g greater than 0 $g\gt0$ . This completes the proof.

Remark 3·3. Note that the ends of Lagrangian cobordisms that we get in Theorem 1·1 are stabilised. One could try to extend Theorem 1·1 to Lagrangian cobordisms with fillable Legendrian ends as Dimitroglou Rizell and the first author did for Lagrangian concordances in [ Reference Dimitroglou Rizell and Golovko9 ]. In [ Reference Dimitroglou Rizell and Golovko9 ], Dimitroglou Rizell and the first author used Legendrian Whitehead doubling to achieve this goal. Unfortunately, it is difficult to mimic the application of Legendrian Whitehead doubling in these settings. The reason for that is that in the proof of Theorem 1·1 we strongly rely on the first part of Proposition 3·2 which allows to control the size of dimension Subscript double struck upper F Sub Subscript p Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis dot right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis $\dim_{\mathbb F_p}H_1(\Sigma_2(\cdot);\; \mathbb F_p)$ . Legendrian Whitehead doubling operation unfortunately does not respect (commute with) the connected sum operation. This makes the control of dimension Subscript double struck upper F Sub Subscript p Baseline upper H 1 left parenthesis normal upper Sigma 2 left parenthesis dot right parenthesis semicolon double struck upper F Subscript p Baseline right parenthesis $\dim_{\mathbb F_p}H_1(\Sigma_2(\cdot);\; \mathbb F_p)$ quite difficult.

Footnotes

This research was supported by by the grants GAČR 26-20231L, GAUK 134125, PRIMUS/24/SCI/009

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Figure 0

Fig. 1. Left: Lagrangian filling of the tb=−1$tb=-1$ unknot; Right: Lagrangian pair-of-pants cobordism obtained from the Lagrangian 1-handle attachment.

Figure 1

Fig. 2. Decomposable Lagrangian concordance Ck$C_{k}$ from the tb=−1$tb=-1$ Legendrian unknot U to the Legendrian representative Λk$\Lambda_{k}$ of the pretzel knot P(3,−3,−k)$P(3,-3,-k)$, where k≥3$k\geq 3$, induced by the ambient surgery along the dashed arc. The front projection of Λk$\Lambda_{k}$ has k−3$k-3$ crossings in the dashed box.

Figure 2

Fig. 3. Lagrangian cobordism of genus one from (S+S−)(Λ′)$(S_+S_-)(\Lambda')$ to Λ′$\Lambda'$ obtained by making an ambient surgery along two dashed arcs.