Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
New hyperbolic knots with convex Upsilon invariants constructed.
problem Constructing knots with convex Upsilon invariants.
method Combinatorial method for (1,1)-knots and connected sum operation. result Infinitely many mutually non-concordant hyperbolic knots with convex Upsilon invariants.
Formula for Upsilon invariant of L-space cable knots derived.
problem Calculating the Upsilon invariant for L-space cable knots.
method Using p,ΥK and ΥTp,q to derive a formula. result Integral values of the Upsilon invariant as a knot concordance invariant.
Paper finds knots with vanishing Upsilon but non-trivial secondary Upsilon.
problem Understanding the Upsilon invariant and its secondary version.
method Constructing infinite families of knots and proving conjectures.
result Secondary Upsilon invariant can be non-trivial even if Upsilon is zero.
New Upsilon invariants rule out stable equivalence of knot complexes.
problem Stable equivalence of knot complexes and its invariants.
method Secondary Upsilon invariants defined by Kim and Livingston.
result Relations between Upsilon invariants do not extend to stable equivalence.
The Upsilon invariant bounds cobordisms between knots and their braid index.
problem Bounding cobordisms between knots and their braid index.
method Using Ozsváth, Stipsicz, and Szabó's Upsilon-invariant.
result Established inductive formulas for the Upsilon invariant of torus knots.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
Study satellite operations on Upsilon invariant of knots.
problem Effect of satellite operations on Upsilon invariant of knots.
method Analyzes satellite operations on Upsilon invariant of Ozsvath-Stipsicz-Szabo.
result Shows independence of knots and their satellites in certain cases.
New knots found with zero Upsilon but nonzero epsilon.
problem Finding knots with specific concordance invariants.
method Constructing new knots with linear independence in the smooth concordance group.
result Found knots with vanishing Upsilon but nonzero epsilon.
The Upsilon invariant helps classify fibered knots and their open book decompositions.
problem Classifying fibered knots and their open book decompositions.
method Using the Ozsváth-Stipsicz-Szabó concordance invariant Upsilon.
result Fibered knots satisfying a specific condition are either unique in their smooth concordance classes or provide counterexamples to the Slice-Ribbon Conjecture.
Study calculates alternating torus knots with small braid index.
problem Determining the alternating number of torus knots with specific braid indices.
method Used the upsilon-invariant and known bounds for braid indices 3 and 4.
result Sharp result for alternating number of torus knots with braid index 4 and less.
Upper bounds for Khovanov width and dealternation number derived for positive braids.
problem Determining bounds for Khovanov width and dealternation number of positive braid links.
method Braid-theoretic technique combined with Upsilon invariant.
result Asymptotically sharp upper bounds for Khovanov width and dealternation number in terms of crossing number.
New knot concordance invariants from cyclic covers of prime power.
problem Constructing new knot concordance invariants.
method Considering m-fold cyclic branched covers with m a prime power.
result Computations of new invariants for some families of knots.
New concordance invariants phi and phi_j are defined and studied.
problem Understanding the relationships between different concordance invariants.
method Defined and analyzed new invariants phi and phi_j, and provided recursive formulas.
result Found infinitely many knots with specific combinations of zero and nonzero phi invariant.
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
Defines a knot genus filtration in smooth concordance group.
problem Understanding the structure of the smooth concordance group.
method Uses Heegaard Floer invariants to define and study the filtration.
result Quotient groups with respect to the filtration are infinitely generated.
Research shows Upsilon function singularity location predicts algebraic knot genus.
problem Determining the genus of cobordisms between algebraic knots.
method Uses the first term of the Puiseux characteristic sequence to find the first singularity of the Upsilon function.
result Better bounds on the genus of cobordisms between algebraic knots than the tau invariant.
New homomorphism from Khovanov homology gives slice genus bounds.
problem Understanding slice genus of knots.
method A 1-parameter family of concordance homomorphisms from Khovanov homology.
result Can prove linear independence of certain knot families.
New knot invariants bound genus and concordance genus.
problem Bounding knot genus and concordance genus.
method Defining secondary Upsilon invariants as piecewise linear functions.
result Secondary invariants detect knots not detected by Upsilon.
New infinite-rank summand found in knot concordance group.
problem Existence of knots with trivial Alexander polynomial and infinite-rank summands.
method Utilized knot Floer homology and the Upsilon invariant.
result Existence of a Z^∞-summand in knot concordance group with trivial Alexander polynomial.
New knot concordance invariants derived from regions in the plane.
problem Knot concordance and distinguishing knots from thin or algebraic ones.
method Associate invariants to regions in the plane, compute for specific knots, and use to obstruct concordances.
result Compute and use new invariants to obstruct concordances to specific types of knots.
Study improves bounds on non-orientable slice genus using knot signatures and concordance invariants.
problem Improving bounds on the non-orientable slice genus of knots.
method Negative surgeries on knots, lower bound derivation using signature and concordance invariants.
result Superadditivity of bounds on stable non-orientable genus, sometimes better than bounds on γ4(K). Study on knot concordance invariant under cabling operations.
problem Behavior of knot concordance invariant under cabling.
method Analyze ΥK(t) and ΥKp,q(t), derive inequalities. result Generalized inequalities for ΥK(t) and ΥKp,q(t). The paper calculates a knot invariant for 3-braid knots.
problem Calculating the concordance invariant for 3-braid knots.
method Constructing cobordisms between 3-braid knots and torus knots.
result Explicit formulas and values for the invariant υ(K) for 3-braid knots. 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.
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.