
Almost positive links are strongly quasipositive. [ArXiv:1809.06692]
With Lukas Lewark and Andrew Lobb.Abstract: We prove that a link is strongly quasipositive if it admits a diagram with a single negative crossing. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive Seifert surfaces produced by Seifert's algorithm.
Approach via surfaces rather than diagrams: We focus on canonical surfaces associated with link diagrams. We perform simplifying operations on diagrams that change the link type(!), but change the canonical surfaces in a `controlled' way. Explicitly, we control quasipositivity of Seifert surfaces by relying on its behaviour under Murasugi summing and taking incompressible surfaces. `Merging of Seifert circles' (depicted) is one such simplifying operation: it reduces the number of Seifert circles (aka vertices in the Seifert graph).
For the proof of the title theorem, this approach is in contrast to a `direct diagrammatical' approach of turning (through link isotopies) a given almost positive diagram into a standard diagram of a strongly quasipositive link.
Combinatorics of plane graphs: For our second main result, we use simplifying operations (e.g. merging of Seifert circles) and the existence of a canonical embedding of the Seifert graph for the relevant link diagrams into the plane (depicted), to reduce to a purely combinatorial problem about bipartite plane graphs. 
Up to topological concordance links are strongly quasipositive.
[ArXiv:1802.02493], in press at J. Math. Pures Appl.
With Maciej Borodzik.Abstract: We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.
Context: Quasipositive links arise in the study of complex plane curves and their study is natural in the context of smooth concordance. In fact, results about complex curves (like the Thom conjecture proven by Kronheimer and Mrowka) places strong restrictions on the smooth concordance classes of links that contain quasipositive links. In contrast, our result establishes that quasipositive links are nondistinguishable from other links through the lens of topological concordance.
The proof: For the proof, we start with a link L, realize it as the boundary of a Seifert surface, which we isotoped to consist of vertical disks and hooked horizontal bands (`hooks') (Figure). Using that a Seifert surface is quasipositive (qp) if all hooks `go up', we iteratively apply a 3D construction to turn the Seifert surface into a qp Seifert surface (in particular, its boundary is a strongly quasipositive link by definition). Finally, we apply the disk embedding theorem to observe that the boundary of the resulting qp Seifert surface is topologically concordant to L.
3D construction: Tying a zeroframed nontrivial knot that is strongly quasipositive into a hook that goes down results in a Seifert surface that can be represented with one less hook that goes down (Figure). The novelty here is that this can be done with every(!) nontrivial strongly quasipositive knot; previously, Rudolph had done this with the righthanded trefoil only to realize all Seifert forms among qp Seifert surfaces.
4D punch line: Restricting the knots that are tied into the hooks to nontrivial strongly quasipositive knots with Alexander polynomial 1 implies (using Freedman's disk embedding theorem) that the 3D construction preserves the topological concordance class of the boundary. 
Calculating the homology and intersection form of a 4manifold from a trisection diagram.
[ArXiv:1711.04762][Proc. Natl. Acad. Sci. USA, Trisections of Smooth Manifolds Special Feature]
With Michael Klug, Trent Schirmer, and Drew Zemke.Abstract: Given a diagram for a trisection of a 4manifold X, we describe the homology and the intersection form of X in terms of the three subgroups of the first homology of a diagram surface S for X. These three subgroups are generated by the three sets of curves and the intersection pairing on S. This includes explicit formulas for the second and third homology groups of X as well as an algorithm to compute the intersection form. Moreover, we show that all (g;k,0,0)trisections admit "algebraically trivial" diagrams.

Braids with as many full twists as strands realize the braid index.
[ArXiv:1708.04998]
With Diana Hubbard.Abstract: We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the functionvalued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that nbraids with fractional Dehn twist coefficient larger than n1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that ntimes twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

On classical upper bounds for slice genera.
[ArXiv:1611.02679]
[Selecta Math.]
With Lukas Lewark.Abstract: We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the fourball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in threespace, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing CassonGordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the Sequivalence class of Seifert matrices.
Classical invariants ... : To a knot one can associate a Seifert matrix, which is welldefined on isotopy classes of knots up to socalled Sequivalence. Knot invariants determined by the Sequivalence classes of Seifert matrices associated with a knot are arguably among the simplest knot invariants. Such knot invariants are said to be `classical'. The bestknown example is the Alexander polynomial of a knot.
... are geometric, ... : More geometrically, two knots have the same classical invariants if and only if the first homology group of the infinite cyclic cover of their complements are isomorphic (as Z[t,1/t]modules together with a natural pairing called the Blanchfield pairing). In other words, this first homology group (with pairing structure) is the `universal' classical invariant. The Alexander polynomial is the order (as Z[t,1/t]modul) of this homology group and a knot has Alexander polynomial 1 if and only if this homology group is trivial.
... correspond to the first interesting term in the derived series ... : Let G denote the fundamental group of the complement of a knot. The isomorphism type of G is essentially (peripheral structure!) a complete knot invariant. By Alexander duality G/[G,G] is infinite cyclic, making [G,G]/[[G,G],[G,G]] (aka the first homology group of the infinite cyclic cover of the knot complement) the first interesting entry in the derived series of G. Of course [G,G]/[[G,G],[G,G]] (with its Blanchfield pairing up to appropriate isomorphy) is the universal classical invariant described above.
... and give upper bounds for the topological 4genus: Surprisingly, for a knot to have the same classical invariants as the unknot (which is equivalent to having Alexander polynomial 1) is equivalent to the following concordance theory statement: the knot arises as the boundary of a locally flat disc in the 4ball such that the fundamental group of its complement is cyclic.* This is a celebrated consequence of Freedman's Disk Embedding Theorem. This article defines the socalled algebraic genus (purely in terms of Sequivalences of Seifert matrices), which bundles earlier classical invariant considerations into a classical invariant that provides a (conjecturally optimal with in classical invariants) upper bound on the topological 4ball genus of knots. The above result by Freedman is a crucial input and can be understood as the `genuszero case' of our work. The diagram illustrates connections of the algebraic genus g_{alg} to other knot invariants. Arrows indicate inequality, dotted arrows are conjectural, and [θ] indicates that an invariant is classical (aka only depends on the Sequivalance class [θ] of Seifert matrices).
* While correct in the first preprint, the unfortunate typo of dropping the fundamental group condition in that statement found its way into the published version; see equation (1). We apologize. 
Khovanov width and dealternation number of positive braid links.
[ArXiv:1610.04534], in press at Math. Res. Lett.
With Sebastian Baader, Lukas Lewark, and Raphael Zentner.Abstract: We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braidtheoretic technique, combined with Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.

Uniqueness of Embeddings of the Affine Line into Algebraic Groups.
[ArXiv:1609.02113], in press at J. Algebraic Geom.
With Immanuel van Santen.Abstract: Let Y be the underlying variety of a connected affine algebraic group. We prove that two embeddings of the affine line C into Y are the same up to an automorphism of Y provided that Y is not isomorphic to a product of a torus (C*)^k and one of the three varieties C^3, SL_2, and PSL_2.
Poster: One page overview PDF of the main result, context, and idea of proof, arranged by Immanuel van Santen for a poster session.

On the Upsilon invariant and satellite knots.
[ArXiv:1604.04901][Math. Z.]
With JungHwan Park and Arunima Ray.Abstract: We study the effect of satellite operations on the Upsilon invariant of OzsvathStipsiczSzabo. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set (D_{2^i,1})_i=1^{infinity} is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive tauinvariant, e.g. the righthanded trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.

On cobordisms between knots, braid index, and the Upsiloninvariant.
[ArXiv:1602.02637]
[Math. Ann.]
With David Krcatovich.Abstract: We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsiloninvariant to provide bounds on cobordisms between knots that `contain fulltwists'. In particular, we recover and generalize a classical consequence of the MortonFranksWilliams inequality for knots: positive braids that contain a positive fulltwist realize the braid index of their closure. We also establish that quasipositive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon invariant of torus knots and compare it to the LevineTristram signature profile.

On the topological 4genus of torus knots.
[ArXiv:1509.07634]
[Trans. Amer. Math. Soc.]
With Sebastian Baader, Lukas Lewark, and Livio Liechti.Abstract: We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with nonmaximal signature invariant.

Alternating numbers of torus knots with small braid index.
[ArXiv:1508.05825]
[Indiana Univ. Math. J.]
With Simon Pohlmann and Raphael Zentner.Abstract: We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsiloninvariant recently introduced by Ozsv\'ath, Stipsicz, and Szab\'o. For the upper bound, we use a known bound for braid index 3 and a new bound for braid index 4. Both bounds coincide, so that we obtain a sharp result.

On 2bridge knots with differing smooth and topological slice genera.
[ArXiv:1508.01431]
[Proc. Amer. Math. Soc.]
With Duncan McCoy.Abstract: We give infinitely many examples of 2bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12crossing knot 12a255. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.

A sharp signature bound for positive fourbraids.
[ArXiv:1508.00418]
[Q. J. Math.]
Abstract: We provide the optimal linear bound for the signature of positive fourbraids in terms of the threegenus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positive threebraids with Gordon and Litherland's approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive fourbraids for which the bounds are sharp are provided.

The degree of the Alexander polynomial is an upper bound for the topological slice genus.
[ArXiv:1504.01064]
[Geom. Topol.]
Abstract: We use the famous knottheoretic consequence of Freedman's disc theorem—knots with trivial Alexander polynomial bound a locallyflat disc in the 4ball—to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice genus.

Optimal Cobordisms between Torus Knots.
[ArXiv:1501.00483]
[Comm. Anal. Geom.]
Abstract: We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in C^2 with the unit 4ball from which a 4ball of smaller radius is removed. Connections to the realization problem of A_nsingularities on algebraic plane curves and the adjacency problem for plane curve singularities are discussed. To obstruct the existence of cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's Upsiloninvariant, which we provide explicitly for torus knots of braid index 3 and 4.

Holomorphically Equivalent Algebraic Embeddings.
[ArXiv:1409.7319],
permanently a preprint.*
With Immanuel van Santen.Abstract: We prove that two algebraic embeddings of a smooth variety X in C^m are the same up to a holomorphic coordinate change, provided that 2dim(X)+1 is smaller than or equal to m. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of C^m.
* Never submitted or planed to be submitted for peerreviewed publication since after a first version of this article appeared on the arXiv, the authors were informed that the main result was previously established by Shulim Kaliman in [ArXiv:1309.3791]/[Proc. Amer. Math. Soc.]. 
Signature and the Alexander polynomial. (An appendix to Livio Liechti's
`Signature, positive Hopf plumbing and the Coxeter transformation.'
[ArXiv:1309.3791]
[Osaka J. Math.])
With Livio Liechti. 
The signature of positive braids is linearly bounded by their first Betti number.
[ArXiv:1401.5336]
[Internat. J. Math.]
Abstract: We provide linear lower bounds for the signature of positive braids in terms of the three genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids.

Gordian adjacency for torus knots.
[ArXiv:1301.5248]
[Algebr. Geom. Topol.]
Abstract: A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using LevineTristram signatures as obstructions to Gordian adjacency. Finally, Gordian adjacency for torus knots is compared to the notion of adjacency for plane curve singularities.