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.
Study shows volume and genus unrelated for hyperbolic fibred knots.
The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now resolved. Two of the cases are settled using invariants of Levine's algebraic concorda…
The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.
New knot invariants from Floer theory help bound knot three-genus.
Study compares nonorientable genus values of torus knots.
Study non-orientable 4-genus for 11-crossing non-alternating knots.
The paper studies triple linking numbers of genus three knots and their derivatives.
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…
Proves any three or more knots can form a genus-zero link in a 3-manifold.
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.
The study computes invariants of satellite knots using bordered Floer homology.
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic v…
The paper generalizes the -genus to characterize slice knots and slice genus.
In this paper, the support genus of all Legendrian right handed trefoil knots and some other Legendrian knots is computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genus with arbitrarily negative Thurston-Benniquin invariant. This answers a…
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…
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, -genus and -genus of a positive knot are equal. In this paper, we p…
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some app…
The paper characterizes Z-slice genus using algebraic unknotting and linking forms.
We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for…
For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link then K is sli…
We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer hom…
We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emph…
Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…
We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.
In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not …
The paper resolves conjectures about knot invariants and shows infinite families of knots.
Study constraints on knot surgery invariants using Seiberg-Witten theory.
We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.
For any alternating knot, it is known that the double branched cover of the -sphere branched over the knot is an -space. We show that the three-fold cyclic branched cover is also an -space for any genus one alternating knot.
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…
New link invariant bounds topological slice genus.
For each three-bridge link of a certain form, we construct a taut Seifert surface for the link and establish whether the link is fibred. Using this, we also give the genus and fibredness of satellite knots whose pattern is constructed from a two-component two-bridge link in the case not addressed by work of Hirasawa an…
We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area…
We define a "reduced" version of the knot Floer complex , and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer -invariants of manifolds arising as surgeries on the knot . As an application to connected sums, we prove that if a knot in the three-sphe…
The paper classifies knots in real projective 3-space and introduces new geometric tools.
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.
The paper sets genus bounds for twisted quantum invariants.
We determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere. Specifically, if surgery along a knot produces a lens space, then there exists an equivalent surgery along a Berge knot with the same knot Floer homology groups. This leads to sharp information about the genus of such a kn…
The study defines and explores almost-concordance classes of knots in 3-manifolds.
We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, w…
This paper classifies knots with simple curves in genus 2 handlebodies.
The study distinguishes knots using finite quotients of their fundamental groups.
Construct divide knots with specific genus properties.
Establishes a rank inequality between knot Floer homologies of freely 2-periodic knots and their quotients.
Classifies knot traces with specific trisection genus limits.
Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satelli…
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…