List of Publications of Marc Kegel
Submitted articles

[23] R. Chatterjee and M. Kegel,
Contact surgery numbers of Sigma(2,3,11) and L(4m+3,4),
 arXiv:2404.18177 (April 2024).
 Abstract: We classify all contact structures with contact surgery number one on the Brieskorn sphere Sigma(2,3,11) with both orientations. We conclude that there exist infinitely many nonisotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact 3sphere. We further prove similar results for some lens spaces: We classify all contact structures with contact surgery number one on lens spaces of the form L(4m+3,4). Along the way, we present an algorithm and a formula for computing the Euler class of a contact structure from a general rational contact surgery description and classify which rational surgeries along Legendrian unknots are tight and which ones are overtwisted.

[22] M. Kegel, A. Ray, J. Spreer, E. Thompson, S. Tillmann,
On a volume invariant of 3manifolds,
 arXiv:2402.04839 (February 2024).
 Abstract: This paper investigates a realvalued topological invariant of 3manifolds called topological volume. For a given 3manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all nonhyperbolic closed 3manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.
 Code and data to accompany this paper can be found
here.

[21] M. Kegel, and N. Weiss,
Complexity of equal 0surgeries,
 arXiv:2401.06015 (January 2024).
 Abstract: We say that two knots are friends if they share the same 0surgery. Two friends with different sliceness status would provide a counterexample to the 4dimensional smooth PoincarĂ© conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4genera. In particular, we compute the minimum of c(K)+c(K') and of t(K)+t(K') among all friends K and K'. Along the way, we classify all 0surgeries of knots of at most 15 crossings.
 Code and data to accompany this paper can be found on our
GitHub page.

[20] M. Kegel, L. Lewark, N. Manikandan, F. Misev, L. Mousseau, and M. Silvero,
On unknotting fibered positive knots and braids,
 arXiv:2312.07339 (December 2023).
 Abstract: The unknotting number u and the genus g of braid positive knots are equal, as shown by Rudolph. We prove the stronger statement that any positive braid diagram of a genus g knot contains g crossings, such that changing them produces a diagram of the trivial knot. Then, we turn to unknotting the more general class of fibered positive knots, for which u=g was conjectured by Stoimenow. We prove that the known ways to unknot braid positive knots do not generalize to fibered positive knots. Namely, we prove that there are fibered positive knots that cannot be unknotted optimally along fibered positive knots; there are fibered positive knots that do not arise as trefoil plumbings; and there are positive diagrams of fibered positive knots of genus g that do not contain g crossings, such that changing them produces a diagram of the trivial knot. In fact, we conjecture that one of our examples is a counterexample to Stoimenow's conjecture.
 Code and data to accompany this paper can be found here.

[19] M. Kegel, N. Manikandan, L. Mousseau, and M. Silvero,
Khovanov homology of positive links and of Lspace 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 Lspace knots have vanishing first Khovanov homology. This suggests a possible extension of our results to Lspace knots.
 Code and data to accompany this paper can be found here.

[18] M. Kegel and F. Schmäschke,
Trisecting a 4dimensional book into three chapters,
 arXiv:2304.12250 (April 2023, revised December 2023).
 Abstract: We describe an algorithm that takes as input an open book decomposition of a closed oriented 4manifold and outputs an explicit trisection diagram of that 4manifold.
Moreover, a slight variation of this algorithm also works for open books on manifolds with nonempty boundary and for 3manifold 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

[17] R. Casals, J. Etnyre, and M. Kegel,
Stein traces and characterizing slopes,
 to appear in Math. Ann., 39 pages.
 arXiv:2111.00265 (November 2021, last revised July 2023).
 Abstract: We show that there exists an infinite family of pairwise nonisotopic Legendrian knots in the standard contact 3sphere 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 3sphere
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 rsurgery, uniquely determines the Legendrian isotopy type.
 Code and data to accompany this paper can be found here.

[16] 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 3manifold can be obtained by rational contact Dehn surgery along a Legendrian link L in the standard tight contact 3sphere.
We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a given contact 3manifold 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 3sphere. 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 nonisotopic contact structures on each of the above manifolds which
cannot be obtained by a single rational contact surgery from the standard tight contact 3sphere. We further obtain results for the 3torus 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.
Published articles (in reverse chronological order)

[15] M. Kegel,
Nonisotopic transverse tori in Engel manifolds,
 Rev. Mat. Iberoam., 40 (2024), 43–56.
 arXiv:2205.04853 (May 2022, revised January 2023).
 Abstract: In every Engel manifold we construct an infinite family of pairwise nonisotopic 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 selflinking number for transverse knots in contact 3manifolds.
Analogous results are presented for Legendrian tori in even contact 4manifolds.

[14] K. Baker and M. Kegel,
Census Lspace knots are braid positive, except for one that is not,
 Algebr. Geom. Topol., 24 (2024), 569–586.
 arXiv:2203.12013 (March 2022, revised April 2022).
 Abstract: We exhibit braid positive presentations for all Lspace 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 1parameter family of hyperbolic Lspace knots that might not be braid positive.
Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic Lspace 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 LiNi.
 Comments: The main article is just 12 pages. The remaining 31 pages is a listing of braid words for the Lspace 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.
 The published version of the braid words of the census knots can be found here.

[13] F. Ayaz, M. Kegel, and K. Mohnke,
The classification of surfaces via normal curves,
 Jahresber. Dtsch. Math.Ver., 125 (2023), 121–134.
 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 3manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.

[12] C. Anderson, K. Baker, X. Gao, M. Kegel, K. Le, K. Miller, S. Onaran, G. Sangston, S. Tripp, A. Wood, and A. Wright,
Lspace knots with tunnel number >1 by experiment,
 Exp. Math., 32 (2023), 600–6014.
 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 Lspace 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 Lspace knot complements.
Furthermore, using SnapPy and KLO we find presentations of these 22 knots as closures of positive braids that realize the MortonFranksWilliams bound on braid index.
The smallest of these has genus 12 and braid index 4.

[11] 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 3manifolds and analyze its basic properties and some of its interesting subgraphs.

[10] 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
S¹orbibundles 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 noncommonly 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 CristofaroGardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3manifolds up to strict contactomorphism.

[9] 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 nullhomologous Legendrian knot as
a count of oriented cusps and linking or intersection numbers; in the case that the manifold has nontrivial 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 nullhomologous Legendrian knot from its projection
to a fixed page.

[8] 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.

[7] S. Durst, H. Geiges, J. Gonzalo, and M. Kegel, Parallelisability of 3manifolds via surgery,
 Expo. Math., 38 (2020), 131–137.
 arXiv:1808.05072 (August 2018).
 Abstract: We present two proofs that all closed, orientable 3manifolds are parallelisable. Both are based on the LickorishWallace surgery
presentation; one proof uses a refinement due to Kaplan and some basic contact geometry. This complements a recent paper by BenedettiLisca.

[6] M. Kegel, J. Schneider, and K. Zehmisch, Symplectic dynamics and the 3sphere,
 Israel J. Math., 235 (2020), 245–254.
 arXiv:1806.08603 (June 2018, revised March 2019).
 Abstract: Given a knot in a closed connected orientable 3manifold we prove that if the exterior of the knot admits an aperiodic contact form
that is Euclidean near the boundary,
then the 3manifold is diffeomorphic to the 3sphere and the knot is the unknot.

[5] 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 d3invariant of the underlying contact structure.

[4] 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 3sphere, i.e. contact surgeries that yield again the standard contact 3sphere.
The main result is that we can exclude nontrivial cosmetic contact surgeries (in sufficiently small tubular neighborhoods) along all transverse knots not isotopic to the transverse unknot with selflinking number 1.
As a corollary it follows that every transverse knot in the standard contact 3sphere is determined by the contactomorphism type of any sufficiently big exterior.
Moreover, we give counterexamples to this for transverse links in the standard contact 3sphere and for transverse knots in general contact manifolds.

[3] S. Durst and M. Kegel, Computing rotation and selflinking numbers in contact surgery diagrams,
 Acta Math. Hungar. 150 (2016), 524–540, (viewonly version),
 Erratum, Acta Math. Hungar. 153 (2017), 537, (viewonly 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 selflinking number of transverse knots. Moreover,
we extend the formula by Ding–Geiges–Stipsicz for computing the d3invariant to (1/n)surgeries.

[2] S. Durst, M. Kegel, and M. Klukas, Computing the Thurston–Bennequin invariant in open books,
 Acta Math. Hungar. 150 (2016), 441–455, (viewonly 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 3manifolds.

[1] 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 3sphere 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 3sphere. On the way a
new userfriendly formula for computing the ThurstonBennequin invariant of a Legendrian knot in a surgery diagram is given.
Nonrefereed contributions

[5] M. Kegel and M. Silvero,
Khovanov homology of positive knots,

[4] M. Kegel, J. Licata, and A. Ray,
Discussions on knot theory in general 3manifolds,
Theses

[3] 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 3sphere are equivalent if and only if their complements are homeomorphic.
By contrast, this does not hold for links in the 3sphere or for knots in general 3manifolds.
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 3sphere 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 3sphere 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.

[2] M. Kegel, Symplektisches Füllen von Torusbündeln (Symplectic Filling of Torus Bundles),
 Masterarbeit (Master thesis), Universität zu Köln (August 2014).
 pdf, 176 pages.

[1] M. Kegel, KontaktDehnChirurgie entlang LegendreKnoten (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.
Coauthors
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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