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Lagrangian Zigzag Cobordisms

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arxiv 2308.02057 v1 pith:MD7ORLB7 submitted 2023-08-03 math.SG math.GT

Joshua M. Sabloff , David Shea Vela-Vick , C.-M. Michael Wong , Angela Wu This is my paper · ORCID
classification math.SGmath.GT
keywords lagrangianconcordancezigzagrelationclassescobordismsformedknots
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abstract

We investigate an equivalence relation on Legendrian knots in the standard contact three-space defined by the existence of an interpolating zigzag of Lagrangian cobordisms. We compare this relation, restricted to genus-$0$ surfaces, to smooth concordance and Lagrangian concordance. We then study the metric monoid formed by the set of Lagrangian zigzag concordance classes, which parallels the metric group formed by the set of smooth concordance classes, proving structural results on torsion and satellite operators. Finally, we discuss the relation of Lagrangian zigzag cobordism to non-classical invariants of Legendrian knots.

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  1. Lagrangian concordance is not a partial order in high dimensions

    math.SG 2024-11 conditional novelty 7.0 of 10

    In R^{4n+1} with n > 1, there exist pairs of non-isotopic loose Legendrian spheres with Lagrangian concordances in both directions, so Lagrangian concordance is not a partial order.

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