List of Publications of Marc Kegel

Submitted articles

• [45] M. Kegel and Q. Ren, Deciding strong quasipositivity and quasipositivity,
  
  • arXiv:2609.17341 (September 2026).
    
  • Abstract: We present two results concerning the decidability of (strong) quasipositivity of links in the 3-sphere. First, using a result of Dynnikov--Prasolov, we provide an algorithm to decide whether or not a link is strongly quasipositive. Second, we reduce the decision problem of the quasipositivity of links to that of braids.
  
  
  
• [44] M. Kegel, S. Y. Li, and Q. Ren, Small undecidable groups and unrecognizable 4-manifolds,
  
  • arXiv:2609.10461 (September 2026).
    
  • Abstract: We construct a 3-generator 9-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with 4 generators and 11 relators, and another with 2 generators and 10 relators. As a consequence, #7(S2×S2) is topologically unrecognizable and #9(S2×S2) is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean 4 formalization of the algebraic results.
  • Comments: The main results were developed by ChatGPT 5.6 Sol.
  • Lean formalization: The Lean formalization is published at Palomar.
  
  
  
• [43] N. M. Dunfield, M. Kegel, S. Y. Li, and Q. Ren, Trisection invariants of 4-manifolds are uncomputable,
  
  • arXiv:2608.27811 (August 2026).
    
  • Abstract: We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.
  
  
  
• [42] C.-S. Hsueh, M. Kegel, D. Suchodoll, and A. Thiele, Legendrian simple knots not detected by Khovanov homology,
  
  • arXiv:2608.24353 (August 2026).
    
  • Abstract: In this short note, we exhibit a Legendrian simple knot that is not detected by Khovanov homology. This disproves a conjecture of Chernov and Maguire.
  
  
  
• [41] M. Kegel and M. Teragaito, An infinite family of strongly invertible L-space knots without Khovanov thin surgery,
  
  • arXiv:2608.00864 (August 2026).
    
  • Abstract: Currently, there exist only three strongly invertible L-space knots that are known to admit no Khovanov thin surgery. In this article, we give the first infinite family of such knots.
  • Code and data to accompany this paper can be found here.
  
  
  
• [40] C. Donatone, M. Kegel, L. Lewark, and P. Truöl, The Conway knot has infinite concordance order,
  
  • arXiv:2606.30422 (June 2026, revised September 2026).
    
  • Abstract: We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.
  • Code and data to accompany this paper can be found here.
  
  
  
• [39] M. Kegel, E. Stenhede, and V. Vértesi, A contact version of Kirby's theorem,
  
  • arXiv:2605.27126 (May 2026).
    
  • Abstract: A theorem of Ding and Geiges states that every closed, connected contact 3-manifold can be obtained from the standard tight contact 3-sphere by contact (+-1)-surgery along a Legendrian link. The literature also contains some examples of contact Kirby moves, i.e. explicit operations on front projections of Legendrian surgery links that change the surgery link but preserve the contactomorphism type of the surgered manifold. Among the most commonly used are cancelling pairs and contact handle slides; however, these moves alone are not sufficient to relate all contact surgery diagrams of contactomorphic contact manifolds.
    In this article, we introduce two new families of contact Kirby moves, called lantern moves and chain moves, and use them to give a complete set of contact Kirby moves. More precisely, we show that two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of planar isotopies, Legendrian Reidemeister moves, insertions or removals of standard cancelling pairs, the two standard contact handle slides, the standard lantern move, and the standard chain move. All these moves are explicit diagrammatic operations in the front projection.
    The proof follows an approach initiated by Avdek through his ribbon-move framework, which is rooted in the Giroux correspondence, and combines it with a presentation by Gervais of the mapping class group. We also discuss several consequences of the main theorem, illustrating the effectiveness of the contact Kirby calculus by recovering the invariance of Gompf's d3-invariant purely diagrammatically and by deriving the topological Kirby theorem from contact-geometric methods.
  
  
  
• [38] M. Kegel and I. Nonino, Transverse knots determined by their cyclic branched covers,
  
  • arXiv:2603.25805 (March 2026).
    
  • Abstract: Harvey-Kawamuro-Plamenevskaya demonstrated the existence of (transversely) non-isotopic transverse knots such that for every n>1 their n-fold cyclic branched covers are contactomorphic. In this short note, we construct other examples of non-isotopic transverse knots that have contactomorphic cyclic branched covers. Conversely, we prove that the transverse isotopy classes of many transverse knots are actually determined by the contactomorphism type of their cyclic branched covers.
  
  
  
• [37] S. Friedl, T. Hirsch, and M. Kegel, The unsolvability of the homeomorphism problem,
  
  • arXiv:2603.23630 (March 2026).
    
  • Abstract: In this short expository note, we give a detailed proof of Markov's theorem on the unsolvability of the homeomorphism problem and of the existence of unrecognizable manifolds in all dimensions larger than 3.
  
  
  
• [36] M. Kegel and J. Spreer, The search for exotic knot traces,
  
  • arXiv:2603.22438 (March 2026).
    
  • Abstract: Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, implemented using SnapPy and Regina, to search for a friend of a given knot. As an application, we construct a census of simple knots that admit friends and use these data to formulate conjectures about knot friends.
  • Code and data to accompany this paper can be found here.
  
  
  
• [35] M. Kegel, I. Nonino, and Monika Yadav, Contact surgery distance,
  
  • arXiv:2512.14904 (December 2025).
    
  • Abstract: In this article, we define the contact surgery distance of two contact 3-manifolds (M,xi) and (M′,xi′) as the minimal number of contact surgeries needed to obtain (M,xi) from (M′,xi′). Our main result states that the contact surgery distance between two contact 3-manifolds is at most 5 larger than the topological surgery distance between the underlying smooth manifolds. As a byproduct of our proof, we classify the rational homology 3-spheres on which the d3-invariant of a 2-plane field already determines its Gamma-invariant and Euler class.
  
  
  
• [34] M. Kegel and M. Schmalian, Links have no characterising slopes,
  
  • arXiv:2511.21506 (November 2025).
    
  • Abstract: We show that there is no analogue of characterising slopes for multi-component links. Concretely, we show that for any ordered link L in S3 with n>1 components and any rational slopes r_1, ..., r_n, there are infinitely many links L_i with non-homeomorphic complements such that the Dehn fillings L(r_1, ..., r_n) and L_i(r_1, ..., r_n) are homeomorphic.
  
  
  
• [33] M. Kegel and M. Schmalian, Unique surgery descriptions along knots,
  
  • arXiv:2508.18521 (August 2025).
    
  • Abstract: We prove that for any non-trivial knot K, infinitely many r-surgeries K(r) along K have a unique surgery description along a knot. Moreover, we show that for any hyperbolic L-space knot K and infinitely many integer slopes n, the manifold K(n) has a unique surgery description. Here we say a 3-manifold M has a unique surgery description along a knot in the 3-sphere if there is a unique pair (K,r) of a knot K and a slope r such that M is orientation-preservingly diffeomorphic to K(r). This generalises the notion of characterising slopes. Conversely, we provide new families of manifolds with several distinct surgery descriptions along knots. More precisely, we construct for every non-zero integer m a knot K_m such that for any integer n, the manifold K_m(m+1/n) can also be obtained by surgery on another knot.
  
  
  
• [32] M. Kegel and L. Piccirillo, Knots that share four surgeries,
  
  • arXiv:2505.13168 (May 2025, revised June 2025).
    
  • Abstract: Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct knots K, K' that have common Dehn surgeries for four distinct slopes. We also construct non-isotopic Legendrian knots K, K' that have contactomorphic contact (+1)- and (-1)-surgeries, disproving an analogous conjecture in contact geometry.
  • Code and data to accompany this paper can be found here.
  
  
  
• [31] 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 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 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.
  
  
  
• [30] 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.
  
  
  

Articles accepted for publication

• [29] V. Bais, A. Di Prisa, D. Hartman, C.-S. Hsueh, M. Kegel, A. Merz, M. Pencovitch, A. Ray, D. Santoro, P. Truöl, L. Wakelin, On the detection of knots by their traces in high dimensions,
  
  • to appear in Proceedings of the London Mathematical Society
  • arXiv:2511.07251 (November 2025, revised September 2026).
    
  • Abstract: For every n>3, we demonstrate the existence of non-isotopic smooth (n−2)-knots in the n-sphere with diffeomorphic traces by generalising the RBG link construction to all dimensions. Conversely, we prove that for every n>3, the unknot in the n-sphere is detected by the diffeomorphism type of its surgery and hence by its trace.
  
  
  
• [28] M. Kegel and P. Truöl, Braid positive surgery descriptions,
  
  • to appear in Glasgow Mathematical Journal
  • DOI: 10.1017/S0017089526100913
  • arXiv:2512.14589 (December 2025, revised March 2026).
    
  • Abstract: In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.
  
  
  
• [27] T. Abe, M. Kegel, and N. Weiss, Complexity of equal 0-surgeries,
  
  • to appear in New York J. Math.
  • arXiv:2401.06015 (January 2024, last revised February 2026).
    
  • Abstract: We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincaré conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. 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 0-surgeries of knots of at most 15 crossings. Moreover, we determine for many friends in our census if their traces are equivalent or not. For that, we develop a new obstruction for two traces being homeomorphic coming from symmetry-exceptional slopes of hyperbolic knots. This is enough to also determine the minimum value of c(K)+c(K') among all friends K and K' whose traces are not homeomorphic.
  • Code and data to accompany this paper can be found on our GitHub page.
  
  
  
• [26] K. Baker, M. Kegel and D. McCoy, Quasi-alternating surgeries,
  
  • to appear in Experimental Math.
  • DOI: 10.1080/10586458.2025.2608234
  • arXiv:2409.09839 (September 2024, revised December 2025).
    
  • Abstract: In this article, we explore phenomena relating to quasi-alternating surgeries on knots, where a quasi-alternating surgery on a knot is a Dehn surgery yielding the double branched cover of a quasi-alternating link. Since the double branched cover of a quasi-alternating link is an L-space, quasi-alternating surgeries are special examples of L-space surgeries.
    We show that all SnapPy census L-space knots admit quasi-alternating surgeries except for the knots t09847 and o9_30634 which both do not have any quasi-alternating surgeries. In particular, this finishes Dunfield's classification of the L-space knots among all SnapPy census knots. In addition, we show that all asymmetric census L-space knots have exactly two quasi-alternating slopes that are consecutive integers. Similar behavior is observed for some of the Baker-Luecke asymmetric L-space knots.
    We also classify the quasi-alternating surgeries on torus knots and explore briefly the notion of formal L-space surgeries. This allows us to give examples of asymmetric formal L-spaces.
  • Code and data to accompany this paper can be found here.
  
  
  
• [25] S. Bastl, R. Burke, R. Chatterjee, S. Dey, A. Durst, S. Friedl, D. Galvin, A. Garcia Rivas, T. Hirsch, C. Hobohm, C.-S. Hsueh, M. Kegel, F. Kern, S. M. S. Lee, C. Löh, N. Manikandan, L. Mousseau, L. Munser, M. Pencovitch, P. Perras, M. Powell, J. P. Quintanilha, L. Schambeck, D. Suchodoll, M. Tancer, A. Thiele, P. Truöl, M. Uschold, S. Vesela, M. Weiß, and M. von Wunsch-Rolshoven, Algorithms in 4-manifold topology,
  
  • to appear in Algebr. Geom. Topol.
  • arXiv:2411.08775 (November 2024, revised September 2025).
    
  • Abstract: We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric.
    In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
  
  
  
• [24] M. Kegel, L. Lewark, N. Manikandan, F. Misev, L. Mousseau, and M. Silvero, On unknotting fibered positive knots and braids,
  
  • to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.
  • DOI: 10.2422/2036-2145.202401_001
  • arXiv:2312.07339 (December 2023, revised September 2024).
    
  • 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.
  
  
  

Published articles

• [23] M. Kegel, A. Ray, J. Spreer, E. Thompson, S. Tillmann, On a volume invariant of 3-manifolds,
  
  • Expo. Math., 44 (2026).
  • DOI: 10.1016/j.exmath.2026.125782
  • arXiv:2402.04839 (February 2024, revised June 2026).
    
  • Abstract: This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold 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 non-hyperbolic closed 3-manifolds 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.
  
  
  
• [22] K. Baker, M. Kegel and D. McCoy, The search for alternating surgeries,
  
  • Journal of Topology, 19 (2026).
  • DOI: 10.1112/topo.70081
  • arXiv:2409.09842 (September 2024, revised May 2026).
    
  • Abstract: Surgery on a knot in the 3-sphere is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of alternating surgery slopes is algorithmically computable and to establish several structural results. Furthermore, we calculate the set of alternating surgery slopes for many examples of knots, including all hyperbolic knots in the SnapPy census. These examples exhibit several interesting phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes. We also establish upper bounds on the set of alternating surgeries, showing that an alternating surgery slope on a hyperbolic knot satisfies |p/q|≤3g(K)+4. Notably, this bound applies to lens space surgeries, thereby strengthening the known genus bounds from the conjecture of Goda and Teragaito.
  • Code and data to accompany this paper can be found here.
  
  
  
• [21] K. Baker, M. Kegel, and K. Motegi, Knots not detected by any trace,
  
  • Int. Math. Res. Not. IMRN., 2026 (2026).
  • DOI: 10.1093/imrn/rnaf375
  • arXiv:2503.14172 (March 2025, revised December 2025).
    
  • Abstract: The first and last named authors have demonstrated the existence of knots for which every integral slope is non-characterizing. In this short note, we extend this result in two ways. There exists a knot that shares for every integer n the same n-trace with infinitely many mutually distinct knots. Moreover, every knot is concordant to a knot that is not detected by any of its traces.
  
  
  
• [20] M. Kegel and M. Yadav, Contact surgery numbers of projective spaces,
  
  • Mediterr. J. Math., 22 (2025).
  • DOI: 10.1007/s00009-025-02936-w
  • arXiv:2504.05032 (April 2025, revised August 2025).
    
  • Abstract: We classify all contact projective spaces with contact surgery number one. In particular, this implies that there exist infinitely many non-isotopic contact structures on the real projective 3-space which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. Large parts of our proofs deal with a detailed analysis of Gompf's Gamma-invariant of tangential 2-plane fields on 3-manifolds. From our main result we also deduce that the Gamma-invariant of a tangential 2-plane field on the real projective 3-space only depends on its d3-invariant.
  
  
  
• [19] K. Baker, M. Kegel and D. McCoy, Two curious strongly invertible L-space knots,
  
  • Adv. Math., 473 (2025), 110287.
  • DOI: 10.1016/j.aim.2025.110287
  • arXiv:2409.09833 (September 2024, last revised April 2025).
    
  • Abstract: We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.
  • Code and data to accompany this paper can be found here.
  
  
  
• [18] M. Kegel and F. Schmäschke, Trisecting a 4-dimensional book into three chapters,
  
  • Geom. Dedicata, 218 (2024), Paper No. 86.
  • DOI: 10.1007/s10711-024-00932-0
  • arXiv:2304.12250 (April 2023, last revised June 2024).
    
  • 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.
  
  
  
• [17] R. Casals, J. Etnyre, and M. Kegel, Stein traces and characterizing slopes,
  
  • Math. Ann., 389 (2024), 1053–1098.
  • DOI: 10.1007/s00208-023-02662-2
  • arXiv:2111.00265 (November 2021, last revised July 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.
  
  
  
• [16] J. Etnyre, M. Kegel, and S. Onaran, Contact surgery numbers,
  
  • J. Symplectic Geom., 21 (2023), 1255–1333.
  • DOI: 10.4310/JSG.2023.v21.n6.a4
  • 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.
  
  
  
• [15] M. Kegel, Non-isotopic transverse tori in Engel manifolds,
  
  • Rev. Mat. Iberoam., 40 (2024), 43–56.
  • DOI: 10.4171/RMI/1413
  • 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.
  
  
  
• [14] K. Baker and M. Kegel, Census L-space knots are braid positive, except for one that is not,
  
  • Algebr. Geom. Topol., 24 (2024), 569–586.
  • DOI: 10.2140/agt.2024.24.569
  • arXiv:2203.12013 (March 2022, last revised August 2026).
    
  • 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. The update in August 2026 added a note that our examples give a negative solution to Problem 1.21(c)(iii) from the K3 Kirby list.
  • 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.
  • DOI: 10.1365/s13291-022-00265-5
  • 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.
  
  
  
• [12] 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,
  
  • Exp. Math., 32 (2023), 600–614.
  • DOI: 10.1080/10586458.2021.1980753
  • 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.
  
  
  
• [11] M. Kegel and S. Onaran, Contact surgery graphs,
  
  • Bull. Aust. Math. Soc., 107 (2023), 146–157.
  • DOI: 10.1017/S0004972722000375
  • 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.
  
  
  
• [10] M. Kegel and C. Lange, A Boothby–Wang theorem for Besse contact manifolds,
  
  • Arnold Math. J., 7 (2021), 225–241.
  • DOI: 10.1007/s40598-020-00165-5
  • 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.
  
  
  
• [9] S. Durst, M. Kegel, and J. Licata, Rotation numbers and the Euler class in open books,
  
  • Michigan Math. J., 70 (2021), 869–888.
  • DOI: 10.1307/mmj/20195780
  • 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.
  
  
  
• [8] S. Durst, H. Geiges, and M. Kegel, Handle homology of manifolds,
  
  • Topology Appl., 256 (2019), 113–127.
  • DOI: 10.1016/j.topol.2019.02.002
  • 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 3-manifolds via surgery,
  
  • Expo. Math., 38 (2020), 131–137.
  • DOI: 10.1016/j.exmath.2018.10.006
  • 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.
  
  
  
• [6] M. Kegel, J. Schneider, and K. Zehmisch, Symplectic dynamics and the 3-sphere,
  
  • Israel J. Math., 235 (2020), 245–254.
  • DOI: 10.1007/s11856-019-1956-5
  • 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.
  
  
  
• [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 d3-invariant 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.
  • DOI: 10.1016/j.topol.2019.01.020
  • 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.
  
  
  
• [3] 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),
  • DOI: 10.1007/s10474-016-0660-8
  • 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.
  
  
  
• [2] 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),
  • DOI: 10.1007/s10474-016-0648-4
  • 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.
  
  
  
• [1] M. Kegel, The Legendrian knot complement problem,
  
  • J. Knot Theory Ramifications 27 (2018), 1850067, 36 pages,
  • DOI: 10.1142/S0218216518500670
  • 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

• [7] M. Kegel, A complete set of contact Kirby moves,
  
  • Oberwolfach Reports, to appear (2026), 3 pages.
  
  
  
• [6] V. Bais, A. Barbensi, B. Burton, D. Celoria, C.-S. Hsueh, M. Kegel, J. Licata, A. Merz, J. Spreer, A. Thiele, and L. Tobin, Models for random knots,
  
  • Oberwolfach Reports 43 (2024), 5 pages.
  
  
  
• [5] M. Kegel and M. Silvero, Khovanov homology of positive knots,
  
  • ICERM Highlights (2023), 3 pages.
  
  
  
• [4] M. Kegel, J. Licata, and A. Ray, Discussions on knot theory in general 3-manifolds,
  
  • Oberwolfach Reports 42 (2021), 4 pages.
  
  
  

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 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.
  
  
  
• [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, 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

• Tetsuya Abe
• Chris Anderson
• Fethi Ayaz
• Valentina Bais
• Kenneth Baker
• Stefan Bastl
• Jan Philipp Bohl
• Rhuaidi Burke
• Roger Casals
• Rima Chatterjee
• Subhankar Dey
• Alessio Di Prisa
• Alison Durst
• Sebastian Durst
• John Etnyre
• Stefan Friedl
• Daniel Galvin
• Xinghua Gao
• Alejandro García Rivas
• Hansjörg Geiges
• Jesús Gonzalo Pérez
• Daniel Hartman
• Tobias Hirsch
• Cara Hobohm
• Chun-Sheng Hsueh
• Frieda Kern
• Mirko Klukas
• Christian Lange
• Khanh Le
• Shun Ming Samuel Lee
• Lukas Lewark
• Joan Licata
• Clara Löh
• Naageswaran Manikandan
• Duncan McCoy
• Alice Merz
• Kyle Miller
• Filip Misev
• Klaus Mohnke
• Monika
• Leo Mousseau
• Kimihiko Motegi
• Lars Munser
• Isacco Nonino
• Sinem Onaran
• Mark Pencovitch
• Patrick Perras
• Lisa Piccirillo
• Mark Powell
• José Pedro Quintanilha
• Arunima Ray
• Geoffrey Sangston
• Diego Santoro
• Lisa Schambeck
• Misha Schmalian
• Felix Schmäschke
• Jay Schneider
• Marithania Silvero
• Jonathan Spreer
• David Suchodoll
• Martin Tancer
• Annika Thiele
• Em Thompson
• Stephan Tillmann
• Samuel Tripp
• Paula Truöl
• Matthias Uschold
• Vera Vertesi
• Simona Veselá
• Laura Wakelin
• Nicolas Weiss
• Melvin Weiß
• Adam Wood
• Ana Wright
• Magdalina von Wunsch-Rolshoven
• Kai Zehmisch
• Daniele Zuddas

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

back