List of Publications of Marc Kegel

Submitted articles

• M. Kegel, N. Manikandan, L. Mousseau, and M. Silvero, Khovanov homology of positive links and of L-space knots,
  
  • arXiv:2304.13613 (April 2023).
    
  • Abstract: We determine the structure of the Khovanov homology groups in homological grading 1 of positive links. More concretely, we show that the first Khovanov homology is supported in a single quantum grading determined by the Seifert genus of the link, where the group is free abelian and of rank determined by the Seifert graph of any of its positive link diagrams. In particular, for a positive link, the first Khovanov homology is vanishing if and only if the link is fibered. Moreover, we extend these results to (p,q)-cables of positive knots whenever q≥p. We also show that several infinite families of Heegaard Floer L-space knots have vanishing first Khovanov homology. This suggests a possible extension of our results to L-space knots.
  • Code and data to accompany this paper can be found here.
  
  
  
• M. Kegel and F. Schmäschke, Trisecting a 4-dimensional book into three chapters,
  
  • arXiv:2304.12250 (April 2023).
    
  • Abstract: We describe an algorithm that takes as input an open book decomposition of a closed oriented 4-manifold and outputs an explicit trisection diagram of that 4-manifold. Moreover, a slight variation of this algorithm also works for open books on manifolds with non-empty boundary and for 3-manifold bundles over the circle. We apply this algorithm to several simple open books, demonstrate that it is compatible with various topological constructions, and argue that it generalizes and unifies several previously known constructions.
  
  
  

Articles accepted for publication

• R. Casals, J. Etnyre, and M. Kegel, Stein traces and characterizing slopes,
  
  • to appear in Math. Ann., 39 pages.
  • arXiv:2111.00265 (November 2021, revised June 2023).
    
  • Abstract: We show that there exists an infinite family of pairwise non-isotopic Legendrian knots in the standard contact 3-sphere whose Stein traces are equivalent. This is the first example of such phenomenon. Different constructions are developed in the article, including a contact annulus twist, explicit Weinstein handlebody equivalences, and a discussion on dualizable patterns in the contact setting. These constructions can be used to systematically construct distinct Legendrian knots in the standard contact 3-sphere with contactomorphic (-1)-surgeries and, in many cases, equivalent Stein traces. In addition, we also discuss characterizing slopes and provide results in the opposite direction, i.e. describe cases in which the Stein trace, or the contactomorphism type of an r-surgery, uniquely determines the Legendrian isotopy type.
  • Code and data to accompany this paper can be found here.
  
  
  
• J. Etnyre, M. Kegel, and S. Onaran, Contact surgery numbers,
  
  • to appear in J. Symplectic Geom., 57 pages.
  • arXiv:2201.00157 (January 2022, revised April 2023).
    
  • Abstract: It is known that any contact 3-manifold can be obtained by rational contact Dehn surgery along a Legendrian link L in the standard tight contact 3-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a given contact 3-manifold under consideration.
    In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three.
    In the second part, we compute contact surgery numbers of all contact structures on the 3-sphere. Moreover, we completely classify the contact structures with contact surgery number one on S¹xS², the Poincaré homology sphere, and the Brieskorn sphere Σ(2,3,7). We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. We further obtain results for the 3-torus and lens spaces.
    As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.
  
  
  
• M. Kegel, Non-isotopic transverse tori in Engel manifolds,
  
  • to appear in Rev. Mat. Iberoam., 10 pages.
  • arXiv:2205.04853 (May 2022, revised January 2023).
    
  • Abstract: In every Engel manifold we construct an infinite family of pairwise non-isotopic transverse tori that are all smoothly isotopic. To distinguish the transverse tori in the family we introduce a homological invariant of transverse tori that is similar to the self-linking number for transverse knots in contact 3-manifolds. Analogous results are presented for Legendrian tori in even contact 4-manifolds.
  
  
  
• F. Ayaz, M. Kegel, and K. Mohnke, The classification of surfaces via normal curves,
  
  • to appear in Jahresber. Dtsch. Math.-Ver., 12 pages.
  • arXiv:2208.00999 (August 2022, revised November 2022).
    
  • Abstract: We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.
  
  
  
• K. Baker and M. Kegel, Census L-space knots are braid positive, except for one that is not,
  
  • to appear in Algebr. Geom. Topol., 45 pages.
  • arXiv:2203.12013 (March 2022, revised April 2022).
    
  • Abstract: We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized is not positive, failing a condition of Ito for braid positive knots.
    We generalize this knot to a 1-parameter family of hyperbolic L-space knots that might not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Furthermore, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li-Ni.
  • Comments: The main article is just 12 pages. The remaining 31 pages is a listing of braid words for the L-space knots in the SnapPy census. Except for o9_30634, these braid words are either positive or negative according to the orientation of the manifold in the census.
  • Code and data to accompany this paper can be found here.
  
  
  
• C. Anderson, K. Baker, X. Gao, M. Kegel, K. Le, K. Miller, S. Onaran, G. Sangston, S. Tripp, A. Wood, and A. Wright, L-space knots with tunnel number >1 by experiment,
  
  • to appear in Exp. Math., 19 pages.
  • arXiv:1909.00790 (September 2019, revised January 2021).
    
  • Abstract: In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in S³, we determine that 22 have tunnel number 2 while the remaining all have tunnel number 1. Notably, these 22 manifolds contain 9 asymmetric L-space knot complements. Furthermore, using SnapPy and KLO we find presentations of these 22 knots as closures of positive braids that realize the Morton-Franks-Williams bound on braid index. The smallest of these has genus 12 and braid index 4.
  
  
  

Published articles (in reverse chronological order)

• M. Kegel and S. Onaran, Contact surgery graphs,
  
  • Bull. Aust. Math. Soc., 107 (2023), 146–157.
  • arXiv:2201.03505 (January 2022).
    
  • Abstract: We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyze its basic properties and some of its interesting subgraphs.
  
  
  
• M. Kegel and C. Lange, A Boothby–Wang theorem for Besse contact manifolds,
  
  • Arnold Math. J., 7 (2021), 225–241.
  • arXiv:2003.10155 (March 2020, last revised July 2020).
    
  • Abstract: A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal S¹-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry. We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result without referring to additional structures. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.
  
  
  
• S. Durst, M. Kegel, and J. Licata, Rotation numbers and the Euler class in open books,
  
  • Michigan Math. J., 70 (2021), 869–888.
  • arXiv:1812.05886 (December 2018, revised January 2020).
    
  • Abstract: This paper introduces techniques for computing a variety of numerical invariants associated to a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of a regular neighborhood of the binding. From this front projection, we compute the rotation number for any null-homologous Legendrian knot as a count of oriented cusps and linking or intersection numbers; in the case that the manifold has non-trivial second homology, we can recover the rotation number with respect to a Seifert surface in any homology class. We also provide explicit formulas for computing the necessary intersection numbers from the front projection, and we compute the Euler class of the contact structure supported by the open book. Finally, we introduce a notion of Lagrangian projection and compute the classical invariants of a null-homologous Legendrian knot from its projection to a fixed page.
  
  
  
• S. Durst, H. Geiges, and M. Kegel, Handle homology of manifolds,
  
  • Topology Appl., 256 (2019), 113–127.
  • arXiv:1811.09055 (November 2018, revised February 2019).
    
  • Abstract: We give an entirely geometric proof, without recourse to cellular homology, of the fact that ∂²=0 in the chain complex defined by a handle decomposition of a given manifold. Topological invariance of the resulting `handle homology' is a consequence of Cerf theory.
  
  
  
• S. Durst, H. Geiges, J. Gonzalo, and M. Kegel, Parallelisability of 3-manifolds via surgery,
  
  • Expo. Math., 38 (2020), 131–137.
  • arXiv:1808.05072 (August 2018).
    
  • Abstract: We present two proofs that all closed, orientable 3-manifolds are parallelisable. Both are based on the Lickorish-Wallace surgery presentation; one proof uses a refinement due to Kaplan and some basic contact geometry. This complements a recent paper by Benedetti-Lisca.
  
  
  
• M. Kegel, J. Schneider, and K. Zehmisch, Symplectic dynamics and the 3-sphere,
  
  • Israel J. Math., 235 (2020), 245–254.
  • arXiv:1806.08603 (June 2018, revised March 2019).
    
  • Abstract: Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot.
  
  
  
• S. Durst and M. Kegel, Computing rotation numbers in open books,
  
  • J. Gökova Geom. Topol. GGT 12 (2018), 71–92,
  • arXiv:1801.01034 (January 2018, last revised January 2019).
    
  • Abstract: We give explicit formulas and algorithms for the computation of the rotation number of a nullhomologous Legendrian knot on a page of a contact open book. On the way, we derive new formulas for the computation of the Thurston–Bennequin invariant of such knots and the Euler class and the d3-invariant of the underlying contact structure.
  
  
  
• M. Kegel, Cosmetic contact surgeries along transverse knots and the knot complement problem,
  
  • Topology Appl., 256 (2019), 46–59.
  • arXiv:1703.05596 (March 2017, revised February 2019).
    
  • Abstract: We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. The main result is that we can exclude non-trivial cosmetic contact surgeries (in sufficiently small tubular neighborhoods) along all transverse knots not isotopic to the transverse unknot with self-linking number -1.
    As a corollary it follows that every transverse knot in the standard contact 3-sphere is determined by the contactomorphism type of any sufficiently big exterior. Moreover, we give counterexamples to this for transverse links in the standard contact 3-sphere and for transverse knots in general contact manifolds.
  
  
  
• S. Durst and M. Kegel, Computing rotation and self-linking numbers in contact surgery diagrams,
  
  • Acta Math. Hungar. 150 (2016), 524–540, (view-only version),
  • Erratum, Acta Math. Hungar. 153 (2017), 537, (view-only version),
  • arXiv:1605.00795 (May 2016, last revised August 2017).
    
  • Abstract: We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding–Geiges–Stipsicz for computing the d3-invariant to (1/n)-surgeries.
  
  
  
• S. Durst, M. Kegel, and M. Klukas, Computing the Thurston–Bennequin invariant in open books,
  
  • Acta Math. Hungar. 150 (2016), 441–455, (view-only version),
  • arXiv:1605.00794 (May 2016, revised July 2016).
    
  • Abstract: We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on contact Heegaard surfaces. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.
  
  
  
• M. Kegel, The Legendrian knot complement problem,
  
  • J. Knot Theory Ramifications 27 (2018), 1850067, 36 pages,
  • arXiv:1604.05196 (April 2016, revised August 2018).
    
  • Abstract: We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3-sphere. On the way a new user-friendly formula for computing the Thurston-Bennequin invariant of a Legendrian knot in a surgery diagram is given.
  
  
  

Non-refereed contributions

• M. Kegel, J. Licata, and A. Ray, Discussions on knot theory in general 3-manifolds,
  
  • Oberwolfach Reports 42 (2021), 4 pages.
  
  
  
• M. Kegel and M. Silvero, Khovanov homology of positive knots,
  
  • ICERM Highlights (2023), 3 pages.
  
  
  

Theses

• M. Kegel, Legendrian knots in surgery diagrams and the knot complement problem,
  
  • Doktorarbeit (Ph.D. thesis), Universität zu Köln (January 2017).
  • pdf, 128 pages. (This version differs slightly from the published version, since I have corrected a few mistake in it.)
  • Abstract: The famous knot complement theorem by Gordon and Luecke states that two knots in the 3-sphere are equivalent if and only if their complements are homeomorphic. By contrast, this does not hold for links in the 3-sphere or for knots in general 3-manifolds.
    In this thesis the same questions are studied for Legendrian and transverse knots and links. The main results are that Legendrian as well as transverse knots in the 3-sphere with the unique tight contact structure are equivalent if and only if their e xteriors are contactomorphic, where the exteriors of the transverse knots have to be sufficiently large. I will also present Legendrian links in the tight contact 3-sphere and Legendrian knots in general contact manifolds that are not determined by their exteriors.
    It will turn out that these questions are closely related to the existence of cosmetic contact surgeries, i.e. contact surgeries that do not change the contact manifold. These are studied with new formulas for computing the classical invariants of Legendrian and transverse knots in the complements of surgery links.
    Another application of cosmetic contact surgeries is that one can switch the crossing type of a given Legendrian or transverse knot by contact surgery without changing the contact manifold. It follows that every Legendrian or transverse knot admits a contact surgery description to an unknot. This yields connections of the classical invariants of Legendrian and transverse knots to the unknotting information of the underlying topological knot type.
  
  
  
• M. Kegel, Symplektisches Füllen von Torusbündeln (Symplectic Filling of Torus Bundles),
  
  • Masterarbeit (Master thesis), Universität zu Köln (August 2014).
  • pdf, 176 pages.
  
  
  
• M. Kegel, Kontakt-Dehn-Chirurgie entlang Legendre-Knoten (Contact Dehn Surgery along Legendrian Knots),
  
  • Bachelorarbeit (Bachelor thesis), Universität zu Köln (September 2011).
  • pdf, 61 pages.
  
  
  

Other documents

• M. Kegel, Negative continued fraction expansions,
  
  • pdf, 3 pages.
  • This short document collects some negative continued fraction expansions which are appear in the context of contact surgery along Legendrian knots.
  
  
  

Co-authors

• Chris Anderson
• Fethi Ayaz
• Kenneth Baker
• Roger Casals
• Sebastian Durst
• John Etnyre
• Xinghua Gao
• Hansjörg Geiges
• Jesús Gonzalo Pérez
• Mirko Klukas
• Christian Lange
• Khanh Le
• Joan Licata
• Naageswaran Manikandan
• Duncan McCoy
• Kyle Miller
• Klaus Mohnke
• Leo Mousseau
• Sinem Onaran
• Geoffrey Sangston
• Felix Schmäschke
• Jay Schneider
• Marithania Silvero
• Samuel Tripp
• Adam Wood
• Ana Wright
• Kai Zehmisch

See also my list of publications on Google Scholar, MathSciNet, zbMATH, Researchgate, and ORCiD, or check my arXiv and Front for the arXiv page.

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