New series invariant for knots and cables, with robustness and relations.
problem Computing series invariants for complex knots and cables.
method Explicit computation and analysis of satellite knots, including a cable of the figure eight knot.
result First example of a cable knot with more than ten crossings, demonstrating robustness and integrality.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
Study compares WRT and CGP invariants using Habiro's series.
problem Comparing WRT and CGP invariants for knots.
method Established relationship between Habiro's series and ADO invariants.
result Difference between WRT and CGP invariants determined by Habiro series.
Formula connects knot complements' invariants.
problem Understanding invariants of knot complements.
method Proposed a connect sum formula for two-variable series invariants.
result Numerical evidence supports the formula for various torus knots.
Innovative series invariant for knot complements, linking to existing invariants.
problem Developing a new series invariant for knot complements.
method Introducing a three-variable series FK(y,z,q) for plumbed knot complements. result Deriving a surgery formula relating FK(y,z,q) to Z^(q) invariant. This review connects knot invariants to quiver representations.
problem Relating knot invariants to quiver representations.
method Relates symmetric quivers and their partition functions to quantum invariants of knots.
result Establishes a correspondence between knot invariants and quiver representations.
Paper connects knot invariants and Morse flow loops.
problem Connecting quantum group invariants and Morse flow loops for knot study.
method Defining a two-variable series invariant by counting Morse flow loops in knot complements and proving it agrees with quantum group BPS series.
result Correspondence proven for all braid-homogeneous knots.
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
problem Understanding topological invariants of hyperbolic 3-manifolds.
method Perturbative power series associated with ideally triangulated cusped hyperbolic 3-manifolds.
result The power series agrees with Kashaev and Andersen-Kashaev invariants to all orders.
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 Bailey pairs derived for tetrahedron index, linking knot invariants.
problem Deriving knot invariants using Bailey pairs for the tetrahedron index.
method Developed new Bailey pairs in terms of q-series.
result New Bailey pairs express the pentagon identity of the tetrahedron index.
The paper generalizes BPS-series for (2,2w+1)-cabling of figure eight knot.
problem Generalizing BPS-series for (2,2w+1)-cabling of figure eight knot. method Verification through recursion method and conjecture analysis.
result Strong evidence for q-holonomic property and formulas for (3,3w+1)-cabling. The paper creates knot invariants using free groups.
problem Invariants of free knots (virtual knots).
method Constructing invariants valued in free groups.
result Series of invariants for free knots.
Study knot invariants to deduce Hopf invariant and propose a slope conjecture.
problem Understanding the topological significance of knot invariants and their relations.
method Analyzing the Gukov-Manolescu knot series and its coefficients, relating to Hopf invariant and colored Jones polynomials.
result Explicit formula for the Hopf invariant in terms of colored Jones polynomials for fibered knots up to 12 crossings.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and q-series to conjecture and compute knot invariants. result Explicit computations match conjectured values for specific knots.
The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in 1/N. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power se…
The physical 3d N=2 theory T[Y] was previously used to predict the existence of some 3-manifold invariants Z^a(q) that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invari…
Invariants of hyperbolic knots connect to quantum modularity.
problem Understanding quantum invariants of hyperbolic knots.
method Introducing matrix invariants and their properties.
result Matrix invariants relate to quantum modularity conjectures.
We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating fun…
New parities defined on virtual knots linked to crossing indices.
problem Defining parities on virtual knots.
method Connecting parities to invariant cycles on arcs and quasi-indices on crossings.
result New series of parities on virtual knots defined.
New formula and properties of inverted Habiro series derived from GM series.
problem Understanding and manipulating knot invariants using series expansions.
method Developed a new formula for the inverted Habiro series (IHS) in terms of GM series and theta functions. Proved a multiplication formula for IHS.
result Established a natural ring structure for IHS and studied its residues, applying them to Dehn surgery formulas.
Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.
problem Quantum invariants of 3-manifolds and links.
method Review of recent developments and connections to other invariants.
result Rich features of quantum invariants like quantum modularity and Verma module structures.
New formulas connect knot invariants with theta functions.
problem Proving conjectures about knot invariants and 3-manifold invariants.
method Inverted state sums and Habiro series.
result Discovered formulas relating knot invariants to theta functions.
We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by term…
Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about kno…
1-loop invariant equals torsion for 2-bridge knots.
problem Proving a conjecture about knot invariants.
method Combining Ohtsuki-Takata work with explicit computation.
result 1-loop term equals Reidemeister torsion for hyperbolic 2-bridge knots.
Study series invariants for plumbed 3-manifolds and their properties.
problem Understanding series invariants for plumbed 3-manifolds and their applications.
method Twisted root lattice, gluing and splitting properties, explicit description of lens spaces and Brieskorn spheres.
result Series verify gluing and splitting properties of 3-manifolds.
The paper is concerned with the Kontsevich-Zagier formal power series f(q)=∑n=0∞(1−q)...(1−qn) and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal power series F(x)=e−1/(24x)f(e−1/x) from which its analytic continuation, i…
The paper studies twisted signature invariants of fibered knots and 3-manifolds.
problem Computing twisted signature invariants of fibered knots and 3-manifolds.
method Reduction to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Use of rings of power series to interpret the twisted Milnor pairing and relate it to twisted Blanchfield pairings.
result New twisted generalizations of the Levine-Tristram signature are derived.
We show that the Vassiliev invariants of orders ≤n of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. As a consequence of this, we obtain that the Vassiliev invariants of …
Study on quantum invariants of twist knots using saddle point method.
problem Asymptotic expansion of Reshetikhin-Turaev invariants of twist knots.
method Saddle point method applied to integral q-surgery. result Asymptotic expansion formula for Reshetikhin-Turaev invariants.
The tail of the colored Jones polynomial of an alternating link is a q-series invariant whose first n terms coincide with the first n terms of the n-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…
The paper defines and computes a knot complement invariant for simple links.
problem Defining and computing a knot complement invariant for simple links.
method Using the large color R-matrix to study the Gukov-Manolescu series.
result Presentation of strange identities for positive braid knots.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Saddle point method applied to colored Jones polynomial.
result Asymptotic expansion formula for twist knots at given root of unity.
The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.
problem Examining the relationship between three-manifold invariants and knot theory.
method Analytic continuation and quiver representation theory.
result Found equalities and patterns in knot theory and quiver representation.
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invari…
We give a formula for the radial asymptotics to all orders of the special q-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in K-theory. The …
We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…
Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and q-series. result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for slN. method Develops symmetrically colored R matrix for slN. result Defines FKslN,sym for positive braid knots. We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…
We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.
In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link L three groups G1,r(L), r>0, G2(L) and G3(L) were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to e…
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2…
For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism invariants rho_n for closed (4k-1)-dimensional manifolds. For 3-dimensional manifo…
The study counts critical points in knot cobordisms using abelian and metacyclic invariants.
problem Counting critical points in knot cobordisms.
method Using homological invariants from cyclic and metacyclic branched covering spaces.
result For each pair of integers g and n, there exists a ribbon knot K with at least n critical points of each index in any genus g cobordism from K to its reverse.
The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of th…