Peter Feller

Besides title, coauthors, abstract, and links to preprint and published versions (if available), some of the items in the list below contain additional informal thoughts that did not get collected elsewhere.
  1. Usually there is a picture here Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space. [ArXiv:1905.08305]
    With Lukas Lewark.

    Abstract: We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimum genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. Using the latter characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched cover of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched cover is determined up to isometry by the action of the deck transformation group on said first homology.


  2. 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.

    Usually there is a picture here 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.


  3. 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 non-distinguishable from other links through the lens of topological concordance.

    Usually there is a picture here 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 zero-framed non-trivial 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(!) non-trivial strongly quasipositive knot; previously, Rudolph had done this with the right-handed trefoil only to realize all Seifert forms among qp Seifert surfaces.

    4D punch line: Restricting the knots that are tied into the hooks to non-trivial 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.

  4. Calculating the homology and intersection form of a 4-manifold 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. Usually there is a picture here

    Abstract: Given a diagram for a trisection of a 4-manifold 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.

  5. Usually there is a picture here Braids with as many full twists as strands realize the braid index. [ArXiv:1708.04998] [J. Topol.]
    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 function-valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n-braids with fractional Dehn twist coefficient larger than n-1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n-times 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.

  6. 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 four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

    Usually there is a picture here Classical invariants ... : To a knot one can associate a Seifert matrix, which is well-defined on isotopy classes of knots up to so-called S-equivalence. Knot invariants determined by the S-equivalence classes of Seifert matrices associated with a knot are arguably among the simplest knot invariants. Such knot invariants are said to be `classical'. The best-known 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 4-genus: 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 4-ball 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 so-called algebraic genus (purely in terms of S-equivalences 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 4-ball genus of knots. The above result by Freedman is a crucial input and can be understood as the `genus-zero case' of our work. The diagram illustrates connections of the algebraic genus galg to other knot invariants. Arrows indicate inequality, dotted arrows are conjectural, and [θ] indicates that an invariant is classical (aka only depends on the S-equivalance 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.

  7. 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 braid-theoretic 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.

  8. 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.

  9. 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 Ozsvath-Stipsicz-Szabo. 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 tau-invariant, e.g. the right-handed 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.

  10. On cobordisms between knots, braid index, and the Upsilon-invariant. [ArXiv:1602.02637] [Math. Ann.]
    With David Krcatovich.

    Abstract: We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive 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 Levine-Tristram signature profile.

  11. On the topological 4-genus 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 non-maximal signature invariant.

  12. 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 upsilon-invariant 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.

  13. On 2-bridge 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 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot 12a255. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.

  14. A sharp signature bound for positive four-braids. [ArXiv:1508.00418] [Q. J. Math.]

    Abstract: We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus 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 three-braids with Gordon and Litherland's approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive four-braids for which the bounds are sharp are provided.

  15. 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 knot-theoretic consequence of Freedman's disc theorem—knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball—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.

  16. 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 4-ball from which a 4-ball of smaller radius is removed. Connections to the realization problem of A_n-singularities 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 Upsilon-invariant, which we provide explicitly for torus knots of braid index 3 and 4.

  17. 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 peer-reviewed 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.].

  18. 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.

  19. 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.

  20. 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 Levine-Tristram signatures as obstructions to Gordian adjacency. Finally, Gordian adjacency for torus knots is compared to the notion of adjacency for plane curve singularities.