Lecture notes on gauge theory for manifold invariants.
problem Constructing invariants of manifolds using gauge theory.
method Gauge-theoretic approach to manifold invariants.
result Introduction to Seiberg-Witten gauge theory.
New surface observables yield 2-knot invariants in nonabelian theories.
problem Developing new invariants for nonabelian theories.
method Introducing surface observables in BF theory and Yang-Mills theory.
result Surface observables induce new 2-knot invariants and electric fluxes.
We introduce a C/Z-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential K-theory, or alternatively, in terms of η-invariants of Dirac operators and local correction terms. In…
New knot invariants from Floer theory help bound knot three-genus.
problem Bounding the three-genus of knots.
method Involutive Heegaard Floer knot theory.
result Two new concordance invariants defined.
Polynomial invariant derived from birack labelling of knots.
problem Developing a polynomial invariant for a broader class of knot theories.
method Generalizing biquandle colouring to birack labelling, reducing to biquandle invariant.
result Polynomial invariant for a class of knot theories.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describ…
Machine learning knot invariants with physics applications.
problem Understanding relations between knot invariants in physics.
method Machine learning and theoretical physics (Chern-Simons theory, gauge theories).
result New analytic results from Big Data experiments.
Study Vassiliev invariants for virtual knots, expanding quantum theory.
problem Understanding Vassiliev invariants for virtual knots.
method Define chord diagrams, weight systems, and Lie algebra weight systems for rotational virtual knots.
result Extended quantum invariants capture more information than standard invariants.
We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…
Real Lie groups' invariant theory matches that of their affine counterparts.
problem Matching invariant theory of real Lie groups with affine groups.
method Simple remark showing coincidence.
result Invariant theory of real Lie groups equals that of their affine counterparts.
The study extends knot theory to knotoids using two approaches.
problem Extending Vassiliev invariants to knotoids.
method Two approaches: 1) Closures to knots, 2) Directly on knotoids.
result Non-trivial type-1 invariants for spherical knotoids.
Khovanov-Floer theories are shown to be invariant under mutation.
problem Invariance of Khovanov-Floer theories under Conway mutation.
method Spectral sequences from Khovanov homology and proofs of conjectures.
result Strong Khovanov-Floer theories are mutation-invariant.
New invariants for 3-manifolds from foliations and noncommutative geometry.
problem Creating new invariants for 3-manifolds.
method Using taut codim-1 foliations and noncommutative geometry techniques.
result Definition of new invariants for 3-manifolds.
Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.
problem Establishing a connection between birational invariants and G-equivariant ones.
method Gromov-Witten theory, Chen-Ruan cohomology, and equivariant atoms.
result New interpretations and applications of classical invariants.
Redefined cord algebra using Morse Theory for knot invariants.
problem Defining a knot invariant using algebraic methods.
method Using Morse Theory to redefine the cord algebra.
result Proved the cord algebra is a knot invariant.
New connections found between knot invariants and Rozansky-Witten theory.
problem Understanding physical interpretations of knot invariants.
method Studying Rozansky-Witten theory with non-compact target spaces.
result New formulations of knot invariants using affine Grassmannians and q-series.
New equivalence found between knot invariants.
problem Understanding relationships between knot invariants.
method Comparing tree reductions of Kontsevich invariant with Orr invariants.
result Orr invariant of degree k is equivalent to tree reduction of Kontsevich invariant of degree <2k.
Proves integrality of knot invariants using number theory.
problem Integrality of BPS invariants of knots.
method Number theoretic proof involving binomial coefficients and Möbius function.
result Proves divisibility properties of knot invariants.
We define a bordism invariant for the fiberwise intersection theory. Under some certain conditions, this invariant is an obstruction for the theory.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
New invariant computed for Brieskorn homology spheres.
problem Computing an invariant for specific manifolds.
method Using Dijkgraaf--Witten invariants in topological K-theory.
result Computed invariant for Brieskorn homology spheres.
The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…
Novel relation between knots and quivers in string theory.
problem Identifying BPS states in string theory.
method Categorification of knot invariants using quivers.
result LMOV invariants of knots can be expressed in terms of motivic Donaldson-Thomas invariants of quivers.
This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gau…
Paper derives localization formula for S1-action and equivariant η-invariants.
problem Localization formula for S1-action and equivariant η-invariants. method Constructing a pre-λ-ring structure in differential K-theory.
result Established a localization formula for equivariant η-invariants.
Formalizes learning algorithm invariances using category theory.
problem Understanding and characterizing invariances in learning algorithms.
method Using category theory to define and formalize invariances of learning algorithms.
result Illustrated and contrasted the invariances of linear regression and ridge regression.
Explains basic knot and link theory for students.
problem Understanding simple invariants of knots and links.
method Defining and explaining linking number, Arf-Casson invariants, Alexander-Conway polynomials, and skein relations.
result Introduction of stronger invariants and the Vassiliev-Kontsevich theorem.
Researchers compute invariants for knots and links in lens spaces using large N and k limits.
problem Computing invariants for knots and links in lens spaces.
method Large N and k limits, saddle point description, collective field theory.
result Invariants for the Hopf link and unknot are computed using on-shell collective field theory action.
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's μ invariant. These invariants are seen to be naturally r…
Diagrammatic method calculates knot invariant related to Chern-Simons theory.
problem Calculating knot invariants in Chern-Simons theory.
method Diagrammatic approach for compact oriented 3-manifolds.
result Diagrammatic method for computing knot invariant d. Floer theories provide invariants for 3- and 4-manifolds.
problem Understanding relationships and expected connections between different Floer theories.
method Comparison and speculation on existing theories and their potential applications.
result Raise questions about the relationship and expected connections between theories.
Paper discusses a new link invariant from virtual link theory.
problem Developing a new invariant for twisted links.
method Using JKSS invariant of virtual links and double coverings.
result Derived a new invariant for twisted links.
Proves equivalence of contact invariants in two Floer homology theories.
problem Identifying contact invariants in different Floer homology theories.
method Proves isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology.
result Identifies contact class with Legendrian invariants in knot Floer homology.
A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki'…
Ihara theory connects braids and links with arithmetic.
problem Understanding relations between braids, links, and arithmetic.
method Investigates analogies between Ihara theory and Galois representations of pure braid groups.
result Established connections between Milnor invariants, Johnson homomorphisms, and Alexander invariants.
New knot invariant Υ defined without holomorphic theory.
problem Defining knot invariants without holomorphic theory.
method Developed grid homology theory to define Υ. result Υ is a well-defined knot invariant and provides a lower bound on unknotting number. Algorithm computes knot invariants for surgeries on prime knots.
problem Computing specific knot invariants for Dehn surgeries.
method Grid homology theory for explicit computation.
result Computed invariants for all prime knots up to 11 crossings.
Study finite type invariants for knots in rational homology 3-spheres.
problem Understanding invariants of knots in rational homology 3-spheres.
method Combinatorial description and analysis of null Lagrangian-preserving surgeries.
result Universal finite type invariants for null-homologous knots with trivial Alexander polynomial.
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…
Unified tau invariants in instanton and monopole Floer theories.
problem Unified tau invariants in instanton and monopole Floer theories.
method Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
result Identified τ_{G} with τ_{G}^{\sharp} via cobordism maps.
Develops invariants for webs and foams using Seiberg-Witten theory.
problem Computing invariants for complex geometric structures.
method Monopole Floer homology with orbifold singularities.
result New invariants for webs and foams.
New invariants derived from ternary quasigroups for knot theory.
problem Developing invariants for knot theory using ternary quasigroups.
method Defined subcomplexes and cocycle invariants based on normalized homology.
result Cocycle invariants defined for ternary quasigroups satisfying specific conditions.
Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…
Survey of quantum enhancements in knot theory.
problem Classifying knots using quantum invariants.
method Collecting and analyzing various quantum invariants.
result New invariants defined by coloring knots with algebraic objects.
Proves smooth torsion invariants match across methods.
problem Matching higher torsion invariants in smooth bundles.
method Explicit comparison of constructions by Igusa/Klein vs. homotopy theory by Badzioch et al.
result Higher torsion invariants agree across methods.
New skein theory for Links-Gould polynomial simplifies link evaluations.
problem Computing Links-Gould polynomial for oriented links.
method Developed a cubic braid-type skein theory.
result Skein theory can evaluate any oriented link.