arXiv research

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…

Lower bounds on rational slice genus using Heegaard Floer invariants.

New lower bound for doubly slice genus using knot signatures.

The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.

New invariants improve Heegaard Floer slice genus and clasp number bounds.

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.

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.

Lower bound on stable 4-genus of knots using Casson-Gordon signatures.

New lower bound for knot genus using Links-Gould invariant.

The paper generalizes the $T$-genus to characterize slice knots and slice genus.

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…

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.

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.

New bounds on genus and area for CMC surfaces in 3-manifolds.

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.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

Local knots can't bound smaller surfaces in rational homology 3-spheres.

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.

Study knots in definite 4-manifolds using minimum-genus bounds.

We prove the compactness of self-shrinkers in $\mathbb R^3$ 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.

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.

New bounds on slice genus from knot invariants.

We prove that if a fibered knot $K$ with genus greater than one in a three-manifold $M$ has a sufficiently complicated monodromy, then $K$ induces a minimal genus Heegaard splitting $P$ that is unique up to isotopy, and small genus Heegaard splittings of $M$ are stabilizations of $P$. We provide a complexity bound in t…

Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.

Study on reducing surgeries on knots, developing thickness and genus bounds.

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.

The paper finds a new lower bound on the genus of surfaces in indefinite 4-manifolds.

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 $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slic…

Optimizes the first eigenvalues of Riemann surfaces for large genus.

Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.

The paper sets genus bounds for twisted quantum invariants.

Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.

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 $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …

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.

New examples show 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…