A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Construct divide knots with specific genus properties.
Study shows four-genus ratio of two-bridge knots decreases as knots get more complex.
Research on the least complex surface in certain 4D shapes.
New examples show clasp numbers can be zero yet four-genus can be arbitrarily large.
We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tri…
The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to the orientable four-ball genus, which is bounded below by the Murasugi signature, the Ozsvath-Szabo tau-invariant, the Rasmussen s-invariant, the best lowe…
Proves partial-dual genus polynomial is a knot invariant weight system.
We calculate the Masur-Veech volume of gothic Teichmüller curves in genus four.
New bounds on nonorientable four-ball genus for torus knots.
Study compares nonorientable genus values of torus knots.
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.
New examples show inequalities can be sharp even when self-linking and genus differ.
Authors show that genus defects of Hopf arborescent links are decidable.
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
Corrects earlier work on surface orbifold pure braid groups.
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general in…
Study non-orientable 4-genus for 11-crossing non-alternating knots.
The study computes invariants of satellite knots using bordered Floer homology.
In this paper we discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumption, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determ…
Formulae count square-tiled surfaces in genus two.
New knots with specific properties have larger 4-genus than previously known bounds.
Two commutators generate mapping class groups of surfaces with genus ≥5.
We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.
Let be two knots with and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a comp…
Extends Rasmussen's invariant to new surfaces in four-manifolds.
Constructs a family of genus three minimal surfaces with parallel ends.
The dihedral genus of a knot is related to its signature and minimal surface genus.
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surf…
New 4-manifolds meet trisection bounds.
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlik…
Paper shows Seiberg-Witten invariants vanish for Davis hyperbolic 4-manifold.
For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.
Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.
We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus and extend these involutions to the four-manifolds obtained by blowing up the …
New examples show algebraically slice knots with specific genus bounds.
We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.
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 fun…
The strong symmetric genus of a finite group is the minimum genus of a compact Riemann surface on which the group acts as a group of automorphisms preserving orientation. A characterization of the infinite number of groups with strong symmetric genus zero and one is well-known and the problem is finite for each strong …
We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…
Non-asphericity of strata of genus-one differentials
In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curv…
The paper proves the existence of maxfaces with multiple swallowtails and planar ends.
Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infin…
We determine the Fukaya Floer homology of the three-manifold which is the product of a Riemann surface of genus times the circle. This sets up the groundwork for finding the structure of the Donaldson invariants of four-manifolds not of simple type in the future. We give the following applications: 1) We show…