New definition of twisted 1-loop invariant using Ptolemy coordinates.
problem Defining and proving properties of twisted 1-loop invariants.
method Alternative definition via Jacobian of Ptolemy coordinates.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial for hyperbolic once-punctured torus bundles.
Twisted Neumann--Zagier matrices for quantum invariants.
problem Constructing quantum invariants from ideal triangulations.
method Define and compute twisted Neumann--Zagier matrices from combinatorics.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial.
Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.
problem Proving a conjecture about polynomial invariants and torsion for fibered 3-manifolds.
method Using combinatorial data of ideal triangulations and layered triangulations of fibered 3-manifolds with toroidal boundary, proving the conjecture for specific cases and confirming it for a large number of nonfibered manifolds.
result The conjecture linking 1-loop invariants to torsion for fibered 3-manifolds with toroidal boundary is proven.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum sl(2;R) and 6j-symbols. result Invariants decay exponentially with hyperbolic volume and 1-loop terms.
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.
New geometric methods solve a conjecture for hyperbolic 3-manifolds.
problem Verifying the 1-loop conjecture for hyperbolic 3-manifolds.
method Constructing geometric ideal triangulations and solving gluing equations.
result Proves the 1-loop conjecture for a large class of hyperbolic 3-manifolds.
New invariants explain topological properties of pseudo-Anosov maps.
problem Understanding invariants of pseudo-Anosov maps at roots of unity.
method Numerical methods and quantum modularity conjecture.
result Descendant invariants related by Fourier transform.
Study topological G₂ and Spin(7) strings at 1-loop using double complexes.
problem Calculate topological string partition functions at 1-loop.
method Define double complexes for supersymmetric backgrounds using generalised geometry, compute partition functions as alternating products of determinants of Laplacians.
result Reproduce known results for G₂ string and predict for Spin(7) string.
Study the kernel of surgery map restricted to 1-loop part of homology cylinders.
problem Understanding the kernel of the surgery map restricted to 1-loop parts of homology cylinders.
method Using Jacobi diagrams and clasper surgery, determine the kernel of the surgery map restricted to the 1-loop part.
result Determined the kernel of the surgery map restricted to the 1-loop part of homology cylinders.
Essential triangulations connect via specific moves in 3-manifolds.
problem Finding essential triangulations in 3-manifolds with infinite boundary components.
method Using 2-3, 3-2, 0-2, and 2-0 moves to connect essential triangulations.
result Essential triangulations are connected via specific moves.
The gluing equations of a cusped hyperbolic 3-manifold M are a system of polynomial equations in the shapes of an ideal triangulation $\calT$ of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann-Zagier datum (comprising the shapes together with the gluing equations in a …
Researchers compute torsion for homology cylinders, proving it a finite-type invariant.
problem Computing finite-type invariants for homology cylinders.
method Non-commutative Reidemeister-Turaev torsion, using K1-group and group ring. result The torsion's reduction is a finite-type invariant of degree d. The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.
problem Analyzing Teichmüller TQFT for hyperbolic knots with generalized FAMED triangulations.
method Introducing generalized FAMED property, proving exponential decay of partition functions in semi-classical limit.
result Partition functions decay exponentially with the volume of knot complements, and the 1-loop invariant emerges.
The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.
problem Understanding the n-loop Kontsevich invariant for knots with identical Alexander polynomials.
method Analyzes the subspace generated by the n-loop Kontsevich invariant of knots with genus ≤ g and same Alexander polynomial.
result For n ≥ 2, the subspace is finite-dimensional.
In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be accounted as a derivation; rather it certifies that an Ansatz found by PSLQ sati…
We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson-Lie T-duality is compatible with the 1-loop renormalization group.
Study proves volume conjecture for specific 3-manifolds.
problem Proving the Andersen-Kashaev volume conjecture for FAMED triangulations.
method Introducing FAMED triangulations and proving existence of Jones function.
result Proves the Andersen-Kashaev volume conjecture for FAMED geometric triangulations.
Study connects surface twists to curve invariants.
problem Understanding fractional Dehn twists on Riemann surfaces.
method Characterization of pseudo-periodic maps and bounds on fractional twists.
result Established relationship between twists and curve invariants.
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate f-twisted Alexander matrices and their connection to quandle cocycle invariants. result Show that f-twisted Alexander invariants of knot quandles are stronger than those of knot groups. Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
problem Understanding and distinguishing twisted links.
method Twisted intersection colorings and double coverings.
result There exist infinitely many pairs of twisted links with equivalent double coverings.
The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.
problem Quantum invariants of knots with additional structure.
method Using twisted Drinfeld doubles and Reshetikhin-Turaev invariants.
result Reidemeister torsion of knot complements as a quantum invariant.
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.
Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.
problem Construct quaternionic-Kähler metrics from special Kähler manifolds with mutually local variations of BPS structures.
method Construct quaternionic-Kähler metrics from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. Provide global and local explicit formulas for the quaternionic-Kähler metric.
result Construct quaternionic-Kähler metrics that are positive-definite and deform the 1-loop corrected Ferrara-Sabharval metric.
We will construct differential forms on the embedding spaces Emb(R^j,R^n) for n-j>=2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions. There are other dimensions in which we can show the closedness if we replace Emb(R^j…
The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.
problem Computing and obstructing the sliceness of knots.
method Algorithmic computation of twisted signature invariants using twisted Blanchfield forms and satellite formulas.
result Illustrated algorithm for (2,q)-torus knots and obstruction of sliceness for certain knots. Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the …
Single twist can unknot certain knots, study shows.
problem Can a knot be unknotted with a single twist?
method Classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
result Obstructions to unknotting with a single twist exist.
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…
Innovative warping labeling for twisted knots and braids.
problem Inventing an invariant for twisted knots and braids.
method Introducing warping degree, constructing warping labeling, extending to virtual braids.
result Developed invariants for twisted knots and braids using warping labeling.
New invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
Invariants for virtual and twisted links using affine indices.
problem Computing invariants for virtual and twisted links.
method Using affine indices to define invariants for virtual and twisted links.
result Invariants for virtual and twisted links computed using affine indices.
This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot K and a representation ρ of the knot group, we define a twisted signature function σK,ρ:S1→Z. This invariant satisfies many of the same algebraic pr…
Study of twisted Alexander matrices for certain quandles and their invariants.
problem Investigate f-twisted Alexander matrices for quandles associated with Alexander pairs. method Define and analyze f-twisted Alexander matrices of certain quandles, relate to Carter-Saito-Satoh's invariant, and discuss connections to quandle homology groups. result 0-th elementary ideal of f-twisted Alexander matrix can be described using Carter-Saito-Satoh's invariant. Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.
problem Deriving sub-Riemannian versions of the Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.
method Derives sub-Riemannian versions of the Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for the twisted BCV spaces.
result Computes Connes conformal invariants for the twisted product and sub-Riemannian limits of these invariants for the twisted BCV spaces.
Study of knot invariants using twisted Iwasawa theory.
problem Determining knot properties like genus and fiberedness.
method Introducing twisted Iwasawa invariants for knot representations and covers.
result Iwasawa invariants determine knot properties and their rigidity.
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.
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, an…
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.
By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. Especially, we get twisted Rokhlin congruences for 8k+4 dimensional spi…
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…
The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.
problem Refining Alexander polynomials and bounds BNSR Σ-invariants for 3-manifolds and Kähler manifolds.
method Introduces twisted homology jump loci and uses tropical geometry to obtain bounds.
result Sharp bounds for BNSR Σ-invariants and obstructions to geometric realizability.
Given a closed symplectic 4-manifold (X,ω), we define a twisted version of the Gromov-Taubes invariants for (X,ω), where the twisting coefficients are induced by the choice of a surface bundle over X. Given a fibered 3-manifold Y, we similarly construct twisted Lefschetz zeta functions associated with surface b…
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidem…
We show that surgery on a connected clover (or clasper) with at least one loop preserves the concordance class of a knot. Surgery on a slightly more special class of clovers preserves invertible concordance. We also show that the converse is false. Similar results hold for clovers with at least two loops vs. S-equivale…
We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form Σ×[0,1] for Σ a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axiom…
For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,E,H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H=∑ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree…