Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
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…
New examples show clasp numbers can be zero yet four-genus can be arbitrarily large.
problem Understanding the relationship between clasp numbers and four-genus for knots.
method Constructing knots with specific clasp numbers and four-genus values.
result Examples of knots with zero clasp numbers but arbitrarily large four-genus.
We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.
The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.
We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let T(p,q) be any torus knot with p even and q odd. The difference between these two invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and k≥0. Hence, the…
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.
Let K1,K2 be two knots with t(K1)+t(K2)>2 and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a comp…
We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, al…
Let D+(K,t) denote the positive t-twisted double of K. For a fixed integer-valued additive concordance invariant ν that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined tν(K) to be the greatest integer t such that $ν(D_+(K,t))…
Study on the cobordism distance between knots and their reverses.
problem Understanding the relationship between a knot and its reverse in terms of cobordism distance.
method Examined the cobordism distance d(K,K^r) for various knots and their properties.
result For knots with g_4(K) = g_3(K), d(K,K^r) is strictly less than twice g_4(K).
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
problem Analyzing Khovanov homology of Turaev genus one links.
method Examined Khovanov homology in specific polynomial gradings.
result Found a trivial summand in Khovanov homology of Turaev genus one links.
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
We study Heegaard Floer homology and various related invariants (such as the h-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the h-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Fl…
The study computes invariants of satellite knots using bordered Floer homology.
problem Computing invariants of satellite knots with specific patterns.
method Using bordered Floer homology and the immersed curve interpretation of the bordered pairing theorem.
result Satellites with thin fibered companions or specific patterns have thin knot Floer homology.
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…
We define a "reduced" version of the knot Floer complex CFK−(K), and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer d-invariants of manifolds arising as surgeries on the knot K. As an application to connected sums, we prove that if a knot in the three-sphe…
We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Frie…
Let K⊂S3 be a Fox p-colored knot and assume K bounds a locally flat surface S⊂B4 over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X→S4. Its branching set is a closed surface embedded in S4 locally flatly away from one singularity whose li…
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
This review explores TDA and TDL beyond persistent homology.
problem Limitations of persistent homology in capturing topological invariants and homotopic evolution.
method Spectral representations, sheaf theory, Mayer topology, interaction topology, differential topology, geometric topology.
result Review of topological tools for various data types.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Novel M-theory approach classifies topological phases of matter.
problem Classifying and understanding topological phases of matter.
method Establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds, identifying topological phases from internal wrapped 3-manifolds.
result Paves a new route toward the classification of topological phases of matter, including fermionic and non-unitary phases.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we…
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Paper introduces a new topological loss for better convergence.
problem Optimizing topological losses for model's desired topological behavior.
method Introduces a new regularized topology-aware loss function.
result Guarantees efficient optimization of the new loss function.
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called…
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
A novel method extracts topological features from word embeddings for text classification.
problem High dimensional and noisy text representations in natural language processing.
method Persistent homology for topological data analysis on word embeddings.
result Topological features outperform conventional text mining features on long textual documents.
Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Topology aids in solving machine learning classification problems.
problem Machine learning classification problems.
method Classical topology applied to neural networks.
result Topology guides neural network architecture and training.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.