In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …
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48 results for “string invariants”
We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant , are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is appli…
Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice virtual open string that is not ribbon. Definitions are extended to open n-strings.
A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connecti…
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to -move and -move.
result Classifies welded string links up to -move and -move.
Multistring based matricesmath.GT
A new method for virtual homotopy classes of virtual strings.
problem Addressing an open problem of Turaev regarding virtual homotopy classes of virtual strings.
method Generalizing Turaev's method to multistring based matrices for virtual n-strings.
result Construction of similar invariants for virtual homotopy classes of virtual n-strings.
Unified method proves integrality of LMOV invariants for framed unknot.
problem Proving the integrality of LMOV invariants for framed unknots.
method Unified method to prove integrality of explicit formulae for LMOV invariants.
result Proves integrality of certain LMOV invariants for framed unknots in higher genera.
Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.
problem Independence of Kontsevich invariant under associator choice for 2-component string links.
method Investigation of Kontsevich invariant for 2-component string links and action of Grothendieck--Teichmüller group.
result Non-trivial action of Grothendieck--Teichmüller group on algebra of 2-component string links.
STRING improves 2D and 3D position encodings for better performance.
problem Efficient and accurate position encoding for 2D and 3D applications.
method STRING extends Rotary Position Encodings with a unifying theoretical framework, maintaining translation invariance and low computational cost.
result STRING shows substantial gains in open-vocabulary object detection and robotics.
Paper shows string cobordism at 24 dims can be determined by elliptic genus.
problem Determining string cobordism at dimension 24.
method Using elliptic genus as an invariant.
result Shows string cobordism at 24 dims can be determined by elliptic genus.
The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.
problem Defining and relating cyclic sets from ribbon string links.
method Endowing ribbon string links with cyclic and cocyclic structures, relating to coend of a ribbon category via quantum invariants.
result Established a relationship between ribbon string links and quantum invariants.
Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.
problem Developing invariants for string links.
method Define and study total Milnor and infinitesimal Morita-Milnor homomorphisms.
result Introduced new invariants for string links.
This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.
problem Finite type invariants for ribbon knotted surfaces and their relation to welded string links.
method Developed a theory of finite type invariants for welded string links up to -equivalence, studied algebraic structures, and showed characterizations.
result Characterizes the information contained by finite type invariants in low degrees for welded string links.
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…
Variant of string topology coproduct invariance formula found.
problem Failure of string topology coproduct invariance under homotopy equivalences.
method Obstruction class built from higher homotopy data and fake diagonal.
result Vanishing of obstruction class measures smallness of homotopy equivalences.
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 invariant. These invariants are seen to be naturally r…
In this note we revisit the subject of anomaly cancelation in string theory and M-theory on manifolds with String structure and give three observations. First, that on String manifolds there is no E8 x E8 global anomaly in heterotic string theory. Second, that the description of the anomaly in the phase of the M-theory…
Virtual stringsmath.GT
A virtual string is a scheme of self-intersections of a closed curve on a surface. We introduce virtual strings and study their geometric properties and homotopy invariants. We also discuss connections between virtual strings, Gauss words, and virtual knots.
String links classified up to -moves and link-homotopy.
problem Classifying string links up to specific moves and link-homotopy.
method Using Milnor link-homotopy invariants and -moves.
result Equivalence classes of string links form a finite group.
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.
We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.
E-string theory reveals modular properties of 4-manifold invariants.
problem Understanding the topology of 4-dimensional manifolds.
method Computing partition function on and verifying its modular properties.
result The partition function of the E-string theory is modular and can be lifted to a topological modular form.
Novel prediction method using string invariants with evolutionary optimization.
problem Optimal prediction parameters for string invariants.
method Evolutionary algorithm for parameter optimization.
result Method performs well in single step prediction but needs improvement for multiple steps.
This paper characterizes Milnor invariants using diagrammatic methods.
problem Classifying links and string links based on their Milnor invariants.
method Diagrammatical characterization using welded knot theory and arrow calculus.
result Conditions for equality of Milnor invariants and triviality of certain lengths.
A virtual string can be defined as a closed curve on a surface modulo certain equivalence relations. Turaev defined several invariants of virtual strings which we use to produce a table of virtual strings up to 4 crossings. We discuss progress in extending the tabulation to 5 crossings. We also provide a counter-exampl…
In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation we introduce the notion of virtual flat biqua…
In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.
We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then…
Proves Stolz conjecture for specific types of manifolds.
problem String manifolds with positive Ricci curvature and Witten genus.
method Analyzes toric string Fano manifolds and string torus manifolds with invariant metrics.
result Proves conjecture for specified types of manifolds.
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of and equivariant concordance groups of strongly invertible knots.
We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …
Study LMOV invariants for a framed unknot in toric Calabi-Yau 3-folds.
problem Computing LMOV invariants for a specific topological string setup.
method Explicit computation and reformulation of LMOV invariants for genus 0 and higher.
result Proved integrality of LMOV invariants and reformulated higher genus invariants.
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We state a computational result on mod-2 finite-type invariants of 2-strand string l…
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
Study String structures using algebraic topology and generalize Witten genera.
problem Understanding algebraic topology aspects of String structures.
method Using Whitehead tower and loop group of , extend generalized Witten genera.
result Vanishing results for generalized Witten genera corresponding to String structures.
The paper analyzes the stability of an observer error in a vibrating string system.
problem Stability analysis of observer error in a vibrating string system.
method Abstract Cauchy problem reformulation and application of LaSalle's invariance principle for infinite-dimensional systems.
result The observer error is asymptotically stable.
New formulas for spatial 2-bouquet graphs discovered.
problem Finding formulas for Vassiliev invariants of spatial 2-bouquet graphs.
method Introducing new Gauss diagram formulas for flat vertex isotopy classes of spatial 2-bouquet graphs.
result First simple example of a Gauss diagram formula for spatial 2-bouquet graphs.
We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway a…
We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a Calabi-Yau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant ver…
Characterizes Milnor invariants with limited repetitions.
problem Understanding Milnor invariants with constraints on repetition.
method Algebraic and diagrammatic approaches using -reduced free groups and welded knot theory.
result Algebraic and diagrammatic characterizations of Milnor invariants.
In this paper we apply a new approach of the string theory to the real financial market. It is direct extension and application of the work [1] into prediction of prices. The models are constructed with an idea of prediction models based on the string invariants (PMBSI). The performance of PMBSI is compared to support …
New Milnor's invariant condition for topologically slice links.
problem Conditions for topologically slice links involving Milnor's invariant.
method Different Milnor's invariant condition to handle cases not covered by previous theorem.
result New sufficient conditions for topologically slice links.
New knots arise from infections by n-component string links that can't be achieved by (n-1)-component infections.
problem Understanding the differences in concordance classes of knots through infections.
method Using satellite constructions and infections to study concordance classes of knots.
result Existence of knots that cannot be obtained by infecting along (n-1)-component string links.
A theory linking invariants, Floer homologies, and Higgs bundles.
problem Understanding complex geometric structures and their invariants.
method Categorification and geometric Langlands correspondence.
result Established a new geometric Langlands correspondence.
We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of…
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in for since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this…
Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…