Positive braid knots with maximal signature have maximal 4-genus.
problem Determining knots with maximal 4-genus.
method Analyzing positive braid knots and their signature invariants.
result All positive braid knots with up to 12 crossings have maximal 4-genus.
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, 4-genus and 3-genus of a positive knot are equal. In this paper, we p…
Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
New bounds on knot genus for positive braids.
problem Understanding the genus difference in positive braids.
method Analyzing the Seifert genus and 4-genus with respect to the number of strands.
result Characterizing knots with specific genus differences by forbidden minors.
The paper finds infinite knots with surfaces of any positive genus.
problem Finding knots with surfaces of arbitrary positive genus.
method Constructing knot exteriors with longitudinal essential surfaces of any positive genus.
result Existence of infinitely many knots with surfaces of any positive genus.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
New open books defy positive factorisation in genus one.
problem Understanding Stein fillable contact structures and their monodromies.
method Constructing an infinite family of open books with specific properties.
result Monodromies of constructed open books do not admit positive factorisations.
The paper classifies knots with a specific bridge position.
problem Identifying knots with a (1,1) decomposition.
method Divided knots into families based on their construction.
result Found three families of knots with a (1,1) decomposition.
Study systole of large genus minimal surfaces in positive Ricci curvature.
problem Understanding the systole of minimal surfaces in manifolds with positive Ricci curvature.
method Colding--Minicozzi lamination theory.
result Results on the systole of large genus minimal surfaces.
Study shows no complete positive scalar curvature for certain 3-manifolds.
problem Existence of complete metrics with positive scalar curvature on specific 3-manifolds.
method Analyzing Whitehead manifold and genus one 3-manifolds.
result No contractible genus one 3-manifold admits a complete metric of positive scalar curvature.
We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinit…
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
We prove several results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard's improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…
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 explores positivity conditions for χy-genus and their implications on Chern numbers and symplectic manifolds.
problem Optimizing Chern number inequalities for almost-complex manifolds.
method Introducing and analyzing positivity conditions for the modified χy-genus and applying them to Chern numbers and symplectic manifolds. result Optimal Chern number inequalities hold for many important Kähler and symplectic manifolds.
We classify genus 2 Lefschetz fibrations on certain 4-manifolds.
problem Classify genus 2 Lefschetz fibrations on specific 4-manifolds.
method Positive factorization of type (10,10) and observations of restrictions.
result Positive factorizations describe genus 2 Lefschetz fibrations on various 4-manifolds.
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.
Handlebody groups are virtual duality groups in positive genus.
problem Understanding the structure of handlebody groups.
method Analyzing mapping class groups and their dualizing modules.
result Description of dualizing modules for handlebody groups.
We create Lefschetz fibrations for knot traces of specific types of knots.
problem Constructing Lefschetz fibrations for knot traces of alternating and extended alternating knots.
method Applying a method from Stein surfaces to knot traces, constructing PALFs with specific properties.
result Knot traces of alternating and extended alternating knots admit Lefschetz fibrations with fibers of small genus.
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.
problem Understanding the equivariant Heegaard genus of reducible 3-manifolds with group actions.
method Thin position theory for 3-dimensional orbifolds to establish bounds on equivariant Heegaard genus.
result Sharp bounds on equivariant Heegaard genus of reducible manifolds, similar to tunnel number results.
Proves a new inequality for certain complex surfaces.
problem Analyzes compact Kähler surfaces with positive scalar curvature.
method Uses properties of holomorphic maps and ruled surfaces.
result Proves a 2-systolic inequality on these surfaces.
Constructed genus zero PALF structures on Akbulut-Yasui plugs.
problem Creating PALF structures on specific 3-manifolds.
method Positive factorization in mapping class groups.
result Described monodromies of PALFs on Akbulut-Yasui plugs.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
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…
Ricci-positive manifolds span the kernel of the A^-genus in rational Spin bordism.
problem Rational Spin bordism classes of Ricci-positive manifolds
method Smooth complete intersections of quadrics
result Ricci-positive manifolds span the kernel of the A^-genus Smooth HP^2 bundle over S^4 with nontrivial A-genus found.
problem Existence of Riemannian metrics with positive sectional curvature.
method Explained existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
result Existence of a smooth HP^2 bundle over S^4 with nontrivial A-genus.
Study mapping class group actions on surface fundamental groups.
problem Understanding group actions on surface fundamental groups.
method Computing image of mapping class group generators as outer automorphisms.
result Computed images of mapping class group generators on fundamental groups.
It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorit…
Entropy norm on surface diffeomorphisms is unbounded.
problem Bounding entropy norm on surface diffeomorphisms.
method Analyzing the group of diffeomorphisms of a closed orientable surface.
result Entropy norm is unbounded on the group of diffeomorphisms of a closed orientable surface.
Classifies 3-braid knots with maximal 4-genus using McCoy's method.
problem Classifying 3-braid knots with maximal 4-genus.
method McCoy's twisting method and Xu normal form.
result Upper bounds for the topological 4-genus of 3-braid knots.
Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.
problem Understanding Shimura subvarieties in Jacobian loci for curves of positive genus.
method Analyzing Galois covers of curves and their Shimura subvarieties under specific numerical conditions.
result The Jacobian locus contains infinitely many Shimura subvarieties of positive dimension for g≤4. Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-systems.
New analysis of crushing surfaces of positive genus impacts triangulation complexity.
problem Crushing surfaces of positive genus can drastically change triangulation topology.
method Detailed analysis of crushing effects on closed essential surfaces of positive genus.
result Proved results about triangulation complexity and JSJ decompositions.
Paper shows a surface with positive genus under a tetrahedron, less than cone's area.
problem Existence of area-minimizing surfaces with positive genus spanning a tetrahedron.
method Triple cover approach, using BV functions and Caccioppoli sets.
result Found a surface of positive genus with area less than a conic surface.
The paper shows knots can have zero support genus in certain spaces.
problem Understanding the support genus of Legendrian knots in specific spaces.
method Using open book decompositions and positive Dehn twists.
result Knots and links can have Legendrian representatives with support genus zero in certain spaces.
Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
problem Maximizing the first positive eigenvalue's multiplicity of the Laplacian.
method Analyzing hyperbolic surfaces of genus 3 and 2, proving the Klein quartic's maximality.
result Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
Classifies definite forms from surgeries on knots with small slice genus.
problem Classifying definite forms from surgeries on knots with specific properties.
method Uses Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
result Classifies positive definite intersection forms for surgeries on knots with slice genus at most 2.
New symplectic surfaces and 4-manifolds created via genus-3 pencils.
problem Creating symplectic surfaces and 4-manifolds homeomorphic but not diffeomorphic.
method Explicit construction of symplectic genus-3 Lefschetz pencils.
result Infinite family of symplectic Calabi-Yau surfaces with b_1=2,3,4.
By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divid…
Defines thin position for 4-manifolds and trisections, finding key relations and diagrams.
problem Defining and analyzing thin position for 4-manifolds and their trisections.
method Introduced width of handle decompositions, defined thin position, and studied trisections.
result Described genus 2g+2 and g+2 trisection diagrams for sphere bundles.
The paper explores positivity and irreducibility in Hurwitz spaces related to differentials of the second kind.
problem Positivity and irreducibility in Hurwitz spaces of certain covers of the projective line.
method Analyzes strata of differentials of the second kind with fixed multiplicities of zeros and poles, and applies this to show positivity and irreducibility in Hurwitz spaces.
result The Hurwitz spaces of degree d, genus g covers of P1 with pure branching at all but possibly one branch point are irreducible under certain conditions. We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.
A specific set of 4g+1 elements is shown to generate the Goeritz group of the genus g+1 Heegaard splitting of a genus g handlebody. These generators are consistent with Powell's proposed generating set for the Goeritz group of the genus g+1 splitting of S^3. There are two proofs: one using purely classical techniques a…