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.
The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.
Two complete knot invariants from diagrams, finite or infinite.
New moves help untangle complex knots.
Polynomial invariant derived from birack labelling of knots.
The study distinguishes knots using finite quotients of their fundamental groups.
The study extends knot theory to knotoids using two approaches.
Category theory generalizes finite type invariants using diagrams systems.
We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial …
Study proves nontrivial knots can't undergo cosmetic surgeries.
Classifies symmetries of knots using group actions and orthogonal representation theory.
The paper calculates Alexander polynomials for knots using finite group representations.
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots…
New method speeds up knot computations in 3D.
This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.
New knot theory module shows torsion-ness in number theory.
Paper discusses groups where twisted Alexander polynomials vanish.
We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral …
Defines a new knot invariant and studies its properties.
Survey of Thurston's impact on knot theory.
The paper explores basic properties of knot skein invariants.
Following Goussarov's paper `Interdependent Modifications of Links and Invariants of Finite Degree' [Topology 37 (1998) 595--602] we describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory turns out to be equivalent to the standard one, it nevertheless has its own share…
We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of …
The paper connects GL-racks to knot coloring invariants.
The paper introduces a new filtration for knot invariants and proves the existence of nontrivial knots.
Quantifies the crossing number of knots based on genus and braid index.
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
It is conjectured that for each knot in , the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a class of small knots that includes 2-bridge knots.
Efficient algorithm computes knot invariants quickly.
We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(s) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a …
A singular knot is an immersed circle in with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing filtration on the infinite dimensional vector space spanned by isotopy classes …
Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.
This paper develops a new homology theory for biquandles and discusses geometric realizations.
This is the second of three papers that refine and extend portions of our earlier preprint, "The depth of a knot tunnel." Together, they rework the entire preprint. The theory of tunnel number 1 knots that we introduced in "The tree of knot tunnels" yields a parameterization in which each tunnel is described uniquely b…
We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized Kontsevitch integral (which is consequently not the universal invariant of finite …
A knot is fertile if it can generate all smaller knots through a specific diagram modification.
In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to…
The study shows a limit on cosmetic surgeries for certain knots.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Study algebraic relations of Vassiliev invariants for families of knots.
Researchers map knot complements using 3d theories and half-index calculations.
Vogel's construction links knot invariants to Lie algebras, revealing new insights.
Special knots with many twists have no certain type of surgery.
New diagonal move simplifies knots and links efficiently.
Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given…
Proves properties of instanton knot Floer homology and connected sum formula.
This work identifies a class of moves on knots which translate to -equivalences of the associated -fold branched cyclic covers, for a fixed and any (with respect to the Goussarov-Habiro filtration.) These moves are applied to give a flexible (if specialised) construction of knots for which the Casson-Walk…
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type invariants of knots and homology 3-spheres fit perfectly into this conception. In …
If the group of a 2-knot group has an abelian normal subgroup of rank which is not finitely generated then either has no minimal Seifert hypersurface or is topologically equivalent to Example 10 of Ralph Fox's``{\it A quick trip through knot theory}".