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.
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.
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.
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.
This paper calculates the non-orientable 4-genus for knots with 10 crossings.
problem Determining the non-orientable 4-genus for knots with a specific number of crossings.
method Calculating the minimal first Betti number of non-orientable surfaces smoothly embedded in a 4-ball with boundary the knot.
result The non-orientable 4-genus for knots with 10 crossings has been calculated.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
Computed the 4-genus for all 12-crossing prime knots.
problem Calculating the 4-genus for all prime knots with 12 or fewer crossings.
method Computed the smooth 4-genera of knots with 12 crossings.
result Completed the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
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.
Study equivariant 4-genus of knots in symmetric 4-manifolds.
problem Understanding equivariant 4-genus of knots in symmetric 4-manifolds.
method Developed techniques for constructing slice disks via equivariant tubing construction.
result Equivariant 4-genus can differ from standard and equivariant 4-genus of 4-manifolds.
Researchers refine the non-orientable 4-genus of torus knots using Batson's surfaces.
problem Finding the minimum non-orientable 4-genus for torus knots. method Developed and analyzed Batson's non-orientable spanning surfaces in B4. result Batson's surfaces minimize the non-orientable 4-genus among certain surfaces. Researchers calculate the non-orientable 4-genus for knots with 8 or 9 crossings.
problem Determining the minimum non-orientable surface complexity for knots with specific crossing numbers.
method Computed the non-orientable 4-genus for knots with 8 or 9 crossings using surface embeddings in 4-ball.
result Computed the non-orientable 4-genus for all knots with 8 or 9 crossings, proving conjectures and providing new slicing number bounds.
Study non-orientable 4-genus for 11-crossing non-alternating knots.
problem Computing the non-orientable 4-genus for specific knots.
method Survey tools and use various techniques to calculate the invariant.
result Calculate non-orientable 4-genus for 11-crossing non-alternating knots.
The paper shows knots with specific properties have smaller 4-genus.
problem Determining knots with specific properties based on their 4-genus.
method Analyzing Thurston norm and fiberedness under zero surgery.
result Knots with smaller 4-genus have Property G.
New knots with specific properties have larger 4-genus than previously known bounds.
problem Finding knots with large 4-genus values.
method Constructing 2-solvable and 2-bipolar knots using specific representations and signatures.
result Knots have 4-genus greater than any given integer g.
Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a…
Study finds new knots that bound Möbius bands in 4D space.
problem Identifying knots that bound Möbius bands in 4D.
method Large-scale computational search with knot invariant obstructions.
result New examples of knots with non-orientable 4-genus equal to 1.
New proof of link 4-genus bounds and satellite link invariants.
problem Determining the 4-genus of links and its behavior under satellite operations.
method New proof using Levine-Tristram signatures and satellite constructions.
result The 4-genus of a link does not increase under certain satellite operations.
Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.
We show that the difference between the genus and the stable topological 4-genus of alternating knots is either zero or at least 1/3.
We prove relative versions of the symplectic capping theorem and sufficiency of Giroux's criterion for Stein fillability and use these to study the 4-genus of knots.
Average signature of 2-bridge knots approximates sqrt(2c/π).
problem Estimating the average signature and 4-genus of 2-bridge knots.
method Developed a model for 2-bridge knot diagrams indexed by crossing number, and used it to derive upper bounds for the average 4-genus.
result Upper bound for the average 4-genus of a 2-bridge knot is 9.75c/log c.
New knots found that are 4-genus minimal.
problem Finding knots with minimal 4-genus.
method Constructing infinitely many amphichiral knots with specific properties.
result Knots with 4-genus minimal for each g>0. Study on nonorientable 4-genus of double twist knots.
problem Determining the nonorientable 4-genus of double twist knots.
method Explicit constructions and obstructions from Donaldson's diagonalization theorem.
result Proved bounds on nonorientable 4-genus for infinite subfamilies of double twist knots.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…
New method to untangle knots using null-homologous twists.
problem Finding the minimum number of twists to convert a knot to the unknot.
method Using null-homologous twists as a generalization of crossing changes.
result The untwisting number is at most twice the surgery description number plus 1.
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…
New findings on knot genera using advanced techniques.
problem Understanding the 4-genus of knots, especially strongly invertible and periodic ones.
method Innovative concordance group invariants, Donaldson's theorem, and g-signature.
result Many new examples showing the equivariant 4-genus is larger than the 4-genus.
Study on nonorientable analogue of Milnor's conjecture for torus knots.
problem Prove a conjecture about the nonorientable 4-genus of torus knots.
method Relied on Ozsváth, Stipsicz, and Szabó's lower bound for γ4 and new closed formulas for the signature of torus knots. result Proved the conjecture for many infinite families of torus knots.
The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the un…
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
Geography problem for nonorientable surfaces bounded by knots.
problem Bounding and computing the nonorientable 4-genus of knots.
method Analysis of existing methods, relationships between Betti number and normal Euler class, exploration of families of torus knots, use of Ozsváth-Szabó d-invariant.
result Improvement on the bound for some knots using the Upsilon invariant.
An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number sn(K), bounding it by signatures, Casson-Gordon invariants, and 4-genus. result Found examples where stabilizing number is less than 4-genus.
The paper uses Heegaard Floer homology to find lower bounds on link genera.
problem Finding lower bounds on the 4-genus of links with vanishing pairwise linking numbers.
method Using the h-function from Heegaard Floer homology to describe genera of surfaces bounded by link components.
result The h-function provides lower bounds for the 4-genus of links, and these bounds are sharp for some L-space links.
Study on knot unknotting numbers and their behavior under connected sums.
problem Behavior of knot unknotting numbers under connected sums.
method Analyzing the band-unknotting number and its sub-additivity properties.
result Infinitely many examples showing unb(K1#K2)<unb(K1)+unb(K2) and unb(K1#K2)<unb(Ki) for i=1,2. New Lissajous-toric knots studied with braid representations.
problem Characterizing and understanding Lissajous-toric knots.
method Investigates braid representations and properties of Lissajous-toric knots.
result Upper bounds for 4-genus and examples of trivial knots.
The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, …
This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based tabl…
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.
Study shows knots can have large genus difference from concordance.
problem Understanding genus differences in knots and surfaces.
method Analyzes the topological 4-genus and minimal genus of bounded surfaces.
result Arbitrarily large genus difference between knots and their concordance.
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.
Study eigenvalues of knot covers to bound knot properties.
problem Bounding knot genera, dealternating number, and alternation number.
method Minimal positive eigenvalues of double branched covers over spanning surfaces.
result Eigenvalue bounds provide estimates for knot properties.
New findings on branched covers of quasipositive links and their L-space properties.
problem Understanding the conditions under which branched covers of quasipositive links are L-spaces.
method Analyzing Alexander polynomials and using properties of cyclic covers.
result Conditions for the L-space property of branched covers of quasipositive links, including specific cases for strongly quasipositive and quasipositive links.
Refines knot defect measurement in 3D and 4D.
problem Measuring how far knots are from being alternating.
method Extends spanning surface defect to 4-ball, making comparisons and proving formulas.
result Connected sum formula proven.
New bounds refine how multivariable signature and nullity change under link cobordisms.
problem Understanding how multivariable signature and nullity change under link cobordisms.
method Refined bounds on multivariable signature and nullity using link cobordisms and 0.5-solvable cobordism. result Multivariable signature and nullity are invariant under 0.5-solvable cobordism. Proves any three or more knots can form a genus-zero link in a 3-manifold.
problem Realizing any finite collection of knots as components of a genus-zero link.
method Proves the realizability of knots as components of genus-zero links in 3-manifolds, controlling pairwise linking numbers.
result Any finite collection of at least three isotopy classes of knots can form a genus-zero link in a 3-manifold, satisfying a specific condition.