Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
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.
The study explores knot invariants using roots of unity.
problem Understanding knot invariants at roots of unity.
method Using Reshetikhin-Turaev method and generalizing ADO invariants.
result Clarified definitions and connections between different invariants.
We prove the ADO invariants are a q-holonomic family and establish recursion relations.
problem Understanding the q-holonomic properties of ADO link invariants. method Proving the ADO invariants are a q-holonomic family and establishing recursion relations. result The ADO invariants for r≥2 are a q-holonomic family, satisfying independent recursion relations. Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in S3. method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
Single-colored ADO-3 invariant matches Links-Gould polynomial for 5-braid closures.
problem Matching ADO-3 invariant with Links-Gould polynomial for specific knot types.
method Proved for closures of 5-braids, conjectured for all knots and links.
result Single-colored ADO-3 invariant equals Links-Gould polynomial for 5-braid closures.
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.
Quantum groups give lower genus bounds for links.
problem Finding lower bounds for Seifert genus of links.
method Using unrolled restricted quantum groups at roots of unity and their invariants.
result ADO link polynomials from quantum groups give genus bounds.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by 2g(K)⋅d(H). New quantum knot invariants derived from Verma modules.
problem Constructing universal quantum knot invariants from Verma modules.
method Defining level N universal invariants from finite quotients of Verma modules over quotient rings.
result Maximal universal invariants for prime N, interpolating Jones and ADO polynomials.
The Witten-Reshetikhin-Turaev invariants extend the Jones polynomials of links in S^3 to invariants of links in 3-manifolds. Similarly, in a preceding paper, the authors constructed two 3-manifold invariants N_r and N^0_r which extend the Akutsu-Deguchi-Ohtsuki invariant of links in S^3 colored by complex numbers to li…
New geometric invariant from disc intersections captures all coloured Jones polynomials.
problem Constructing a universal knot invariant from configuration spaces.
method Defining a new local system and Lagrangian submanifolds in the disc.
result The new invariant recovers Habiro's universal invariant and more.
Holonomy invariants from SL2(C) link complements detect link geometry.
problem Detecting geometric information about links using algebraic quantum invariants.
method Enhanced RT construction with SL2(C) holonomy representations. result Holonomy invariants JN compute Reidemeister torsion for N=2.