Knots in 4D manifolds are trivial based on Wedderburn's Theorem.
problem Understanding knots in 4-dimensional manifolds.
method Application of Wedderburn's Theorem.
result All knots and links in smooth 4D manifolds are trivial.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,181 papers · 148 categories
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48 results for “4D knot theory”
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
New invariant measures complexity of 2-knots in 4D space.
problem Measuring complexity of 2-knots in 4D space.
method Introduced shadow-complexity based on Turaev shadows.
result Characterized 2-knots with shadow-complexity up to 1.
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in .
method Develops a theorem for ribbon torus-links in .
result First step towards a 4D Markov theorem.
Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in .
method Constructing surface germs linked to knots in and studying their Lipschitz geometry.
result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
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.
Lecture notes on link homologies and knotted surfaces, focusing on 4D obstructions.
problem Understanding knotted surfaces using link homology theories.
method Overview of link homology theories, introduction to Khovanov homology, and computational techniques.
result New insights into the homomorphisms assigned to link cobordisms for knotted surfaces.
Disproves a conjecture about knots in 4D space.
problem Non-orientable slice genus of torus knots
method Found a specific knot that bounds a Möbius band in 4-ball
result Counterexample to Batson's conjecture
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
New invariant for surface-knots in 4D from marked graphs.
problem Invariants for surface-knots in 4D.
method Marked graph diagrams for surface-knots.
result Invariant constructed for surface-knots.
New invariant for surface-knots in 4D space.
problem Invariants for surface-knots in 4D space.
method Diagrammatic approach in 3D space.
result Invariant constructed for surface-knots.
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
Minimal area of spun trefoil knot is found in 4D cubical space.
problem Finding the minimum area for a spun trefoil knot in 4D cubical space.
method Defined minimal area of cubical 2-knots, used isotopy to find minimum area for spun trefoil.
result Spun trefoil knot requires a specific minimal area in 4D cubical space.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
Study shows how concordance surgery impacts a 4D knot invariant.
problem Understanding how concordance surgery affects a specific 4D knot invariant.
method Used sutured Floer TQFT and a perturbed version of sutured Floer homology.
result Formula involving the graded Lefschetz number of the concordance map on knot Floer homology.
New condition for Alexander polynomial factorization in 4D knots.
problem Alexander polynomial factorization for 2-knots in .
method Alternative notion of ribbon 2-knots to prove factorization condition.
result Topological condition for factorization of Alexander polynomial.
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.
Unified framework for 3D and 4D manifold and knot theory.
problem Unified understanding of manifold and knot theory.
method Unified correspondence between different subfields of low-dimensional topology.
result Unified algebraic manifestations of 3D and 4D manifold and knot theory.
The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.
problem Defining and understanding Murasugi sum in 4D for knotted surfaces.
method Introduced a 4D Murasugi sum to define arborescent knotted surfaces.
result Defined and studied arborescent knotted surfaces using 4D Murasugi sum.
New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
New method finds non-orientable knotted surfaces in 4D.
problem Reducibility of knotted surfaces in 4D.
method Elementary obstruction to reducibility.
result Construction of stably irreducible non-orientable surfaces.
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
The paper introduces new inequalities for knots in 4D cobordisms.
problem Understanding constraints on smooth cobordisms between knots.
method Establishes relative adjunction inequalities using Heegaard Floer homology.
result Produces concordance invariants for knots and links.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
Stably slice disks of linksmath.GT
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number , bounding it by signatures, Casson-Gordon invariants, and 4-genus.
result Found examples where stabilizing number is less than 4-genus.
New invariant measures knotted surfaces in 4D, revealing unknottedness.
problem Measuring knotted surfaces in 4D.
method Defined an integer invariant for bridge trisections of surfaces in or .
result Invariant implies the surface is unknotted.
We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example,…
A class of 3d supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class…
Geometric interpretation of 2d-4d wall-crossing formulas.
problem Understanding wall-crossing phenomena in coupled 2d-4d systems.
method Deformation theory of holomorphic pairs and relation to scattering diagrams.
result Geometric interpretation of wall-crossing formulas.
Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theoryhep-th
New insights into 4d YM and 5d topological field theories with higher symmetries.
problem Exploring new topological field theories with higher symmetries.
method Dynamic gauging of 1-form symmetry, higher anomalies, and lattice simplicial complex regularizations.
result Discovery of new higher-form gauge fields and exotic anyonic statistics.
Cubulated moves for 2-knotsmath.GT
Paper proves isotopic 2-knots can be transformed by cubulated moves.
problem Determining isotopy of 2-knots in 4D.
method Finite sequence of cubulated moves analogous to Reidemeister and Roseman moves.
result Two cubical links in 4D are isotopic if and only if one can pass to the other by cubulated moves.
Study Higgs bundles for M-theory on G2-manifolds.
problem Understanding gauge theories coupled to gravity in M-theory compactifications.
method Derived BPS equations and massless spectrum using Morse and Morse-Bott theories.
result Determined Higgs bundle for TCS G2-manifolds and provided a prescription for chiral matter.
Sharp gap theorem found for Yang-Mills connections in 4D.
problem Yang-Mills connections in 4D.
method Exploiting an associated Yamabe-type problem.
result Sharp conformally invariant gap theorem proved.
Nearly Kaehler Reductionhep-th
We consider compactification of type IIA supergravity on nearly Kaehler manifolds. These represent a simple class of SU(3) structure manifolds which includes S^6 and CP^3. We exhibit for the first time an explicit reduction ansatz in this context, obtaining an N=2 gauged supergravity in 4d with a single vector and hype…
New method shows nonorientable surfaces in 4D are topologically unknotted.
problem Tackles isotopy classes of nonorientable surfaces in .
method Calculations implemented in Sage to show ambient isotopy.
result Closed, nonorientable surfaces in are topologically unknotted.
Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
Ribbon homology cobordismsmath.GT
Study 4D homology cobordisms without 3-handles, deriving obstructions.
problem Understanding 4D homology cobordisms without 3-handles.
method Utilizing Thurston geometries, character varieties, and homologies, derive obstructions.
result Generalize knot Floer homology obstructions for ribbon concordances.
Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.
problem Defining topologically twisted partition functions for 4d N=2 theories.
method Topological twisting for general 4d N=2 theories, introducing generalized spin-c structures.
result Topological partition functions depend on spacetime type, gerbe connections, and generalized spin-c structures.
New classes from 4D gauge theories.
problem Constructing characteristic classes for 4-manifold bundles.
method Using -Yang-Mills theory and Seiberg-Witten theory for families.
result Characteristic classes of 4-manifold bundles constructed.
Fibered knots and disks extend fibrations in 4D space.
problem Extending fibrations from knot complements to disk complements.
method Constructing movies of singular fibrations and identifying sufficient conditions.
result Fibrations extend when disks have exactly two local minima.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point parti…
Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space - hoops can be composed as in π_1, balloons as in π_2, and hoops "act" on balloons as π_1 acts on π_2. We observe that ordinary …
New -holonomy manifolds from 5d N=1 theories domain walls.
problem Geometrizing domain walls in 5d N=1 theories.
method Constructing 7-manifolds by fibering a Calabi-Yau over a real line.
result 7-manifolds with -holonomy from domain walls in 5d theories.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
String theory connects lattice models, links, and geometric Langlands.
problem Connecting lattice models, links, and geometric Langlands.
method T-duality and worldvolume theories in string theory.
result Unified understanding of various mathematical concepts.