In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both…
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48 results for “Genus bounds”
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
New lower bound for doubly slice genus using knot signatures.
problem Finding a lower bound for the doubly slice genus of knots.
method Using the classical signature function to derive a new lower bound.
result Proved that for every nonnegative integer N, there exists a knot with exactly N difference between slice and doubly slice genus.
A note on surfaces in and math.GT
The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.
problem Investigating the genus of surfaces in complex projective spaces.
method Analyzing knots and torus knots in and .
result The -genus of knots is unbounded, unlike its topological counterpart.
New invariants improve Heegaard Floer slice genus and clasp number bounds.
problem Improving bounds for knot concordance.
method Using knot Floer homology and involutive correction terms.
result Improved slice genus and clasp number bounds proved.
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 paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.
problem Classifying group-actions on surfaces of small genus, particularly focusing on bounding and geometrically bounding cases.
method Analyzing large group-actions on surfaces of genus 3, distinguishing between bounding and geometrically bounding cases.
result Identifies which large group-actions on surfaces of genus 3 are bounding or geometrically bounding.
Lower bound on stable 4-genus of knots using Casson-Gordon signatures.
problem Finding a lower bound on the stable 4-genus of knots.
method Using Casson-Gordon τ-signatures to compute the lower bound.
result A twist knot is torsion in the knot concordance group if and only if it has vanishing stable 4-genus.
New lower bound for knot genus using Links-Gould invariant.
problem Finding a tighter lower bound for knot genus.
method Representation theory of to prove degree of Links-Gould polynomial bounds Seifert genus.
result The Links-Gould polynomial provides a new lower bound on knot genus, detecting specific knots like Kinoshita-Terasaka and Conway.
The paper generalizes the -genus to characterize slice knots and slice genus.
problem Characterizing slice knots and slice genus using the -genus.
method Generalizing the -genus to provide a -dimensional characterization of the slice genus.
result The difference between the -genus and the slice genus can be arbitrarily large.
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…
The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.
problem Understanding the complexity of Heegaard splittings induced by fibered knots.
method Analyzing the monodromy of fibered knots and their impact on Heegaard splittings.
result Minimal genus Heegaard splittings of a three-manifold are unique and can be induced by fibered knots with complex monodromies.
A Seifert surface for a knot K is called canonical if it can be built by applying Seifert's algorithm to some projection of K. The canonical genus of K is the smallest genus of a surface so obtained. In this paper we show that there is a bound on the volume of a hyperbolic knot which admits a canonical surface of genus…
Paper bounds the A-hat genus using curvature and isoperimetric constants.
problem Bounding the A-hat genus of Riemannian manifolds.
method Spectral analysis of the Dirac operator, scalar curvature lower bounds, and isoperimetric constants.
result Proves an upper bound on the A-hat genus using manifold properties.
Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.
problem Measuring the minimum genus of non-orientable surfaces bounded by torus knots.
method Computed bounds and provided a generalized formula for non-orientable 4-genus of torus knots.
result Computed bounds and a generalized formula for non-orientable 4-genus of torus knots.
New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
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…
Study shows concordance invariants bound Turaev genus.
problem Understanding the Turaev genus of knots.
method Using differences between concordance invariants, including Rasmussen's -invariant and -invariants.
result Established lower bounds for Turaev genus and provided examples of quasi-alternating knots with specific genus values.
The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.
problem Lowering the rational genus of knots in rational homology 3-spheres.
method Using Heegaard Floer homology and the -invariant.
result Same lower bound and minimizers as Ni and the first author's results.
Local knots can't bound smaller surfaces in rational homology 3-spheres.
problem Understanding local knots and their bounds in rational homology 3-spheres.
method Using Heegaard Floer invariant ν+ and additivity results.
result Local knots from ν+ -sharp knots have rational slice genus equal to the slice genus of the original knot.
In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C^2. To construct su…
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
Study knots in definite 4-manifolds using minimum-genus bounds.
problem Determining whether knots are smoothly slice.
method Minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds, gauge-theoretic obstructions.
result Alternate proof that (2,1)-cable of figure eight knot is not smoothly slice.
We prove the compactness of self-shrinkers in with bounded entropy and fixed genus. As a corollary, we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus.
The paper calculates genus bounds for multibranched surfaces.
problem Finding genus bounds for multibranched surfaces.
method Using the first Betti number and boundary genus, the paper provides lower bounds for maximum and minimum genus.
result The maximum and minimum genus of equals twice that of .
We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound is based on an analysis of broken surface diagrams and embedding properties of 3-…
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…
Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.
problem Understanding CMC hypersurfaces with bounded index and area.
method Bubble-compactness theory for embedded CMC hypersurfaces in low dimensions.
result Minimal blow-ups are all catenoids, and bounds on genus provided.
New bounds on slice genus from knot invariants.
problem Bounding slice genus of knots in .
method Using -invariant to establish lower bounds.
result Proves conjecture on slice genus bounds.
Study on reducing surgeries on knots, developing thickness and genus bounds.
problem Understanding reducible surgeries on knots in .
method Developed thickness bounds for L-space knots and lower bounds on slice genus; used -invariants and mapping cone formula from Heegaard Floer homology.
result Provided new upper bounds on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots; verified the Cabling Conjecture for thin knots.
Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.
problem Understanding the Kähler-hyperbolicity of bounded symmetric domains.
method Defines Kähler-hyperbolicity length by rank and genus, and characterizes it through a special Bergman potential.
result Establishes a unique constant for Kähler-hyperbolicity based on gradient length of a Bergman potential.
On the slice genus of linksmath.GT
We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding …
New bounds on nonorientable four-ball genus for torus knots.
problem Finding bounds on the nonorientable four-ball genus of torus knots.
method Combining knot Floer homology techniques to derive lower bounds.
result Sharp bounds for several families of torus knots, including .
The paper finds a new lower bound on the genus of surfaces in indefinite 4-manifolds.
problem Finding a new lower bound on the genus of surfaces in indefinite 4-manifolds.
method Proves a new lower bound on the genus of a properly embedded surface in representing a given homology class and with boundary a quasipositive knot .
result The minimal genus of such a surface is equal to the slice genus of in the null-homologous case.
We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.
This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot is bounded above by the sum of the slice genera of and . Our main result establishes this conjecture for a variant of the topological slice genus, the -slic…
Optimizes the first eigenvalues of Riemann surfaces for large genus.
problem Finding optimal lower bounds for first eigenvalues of Riemann surfaces.
method Analyzing shortest multi-closed curves to establish a new lower bound.
result The first eigenvalue of a Riemann surface is greater than a specific formula involving the genus and a constant.
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by .
Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.
problem Minimal genus of second homology classes in right angled Artin groups.
method Lower bounds, characterizations, and examples to show minimal genus.
result Minimal genus is half the rank for complete graphs, trees, and complete bipartite graphs, and can be realized by disjoint unions of tori.
We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov -invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …
Lower Bound For Dilatationsmath.GT
We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.
We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus …
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
New examples show algebraically slice knots with specific genus bounds.
problem Understanding slice genus of algebraic knots and their mirrors.
method Genus bound from Casson-Gordon invariants and cabling formula.
result Examples of algebraically slice knots with specific genus bounds.
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…