New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
Paper shows how to twist knots to make them trivial or non-trivial.
problem Understanding the behavior of knots under twist-spinning operations.
method Analyzes the effect of m1- and m2-twist-spinning on classical knots. result Shows conditions for a knot to be trivial or non-trivial after certain twist-spinning operations.
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
Study lattice paths from twist knots and double twist knots.
problem Understanding combinatorics of twist knots and double twist knots.
method Analyzing quiver generating series of HOMFLY-PT polynomial limits.
result Lattice path models for twist knots and double twist knots.
A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in F, the genus 2 Heegaard surface for S3. Primitive/primitive and primitive/Seifert knots lie in F in a particular way. Dean gives sufficient conditions for the parameters of the tw…
Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.
problem Classifying twisted torus knots with Horadam parameters.
method Using recursive Horadam parameters to generalize Lee's work on Fibonacci parameters.
result Families of twisted torus knots with Horadam parameters are provided.
Study finds infinite non-fibered twisted torus knots.
problem Identifying non-fibered twisted torus knots.
method Explicit formula for Alexander polynomial, leading coefficients analysis.
result Infinite families of non-fibered twisted torus knots found.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
Positive braids with at least two twists form hyperbolic knots.
problem Classifying knots formed by specific braids.
method Conditions on positive braids with at least two full twists.
result Closure of such braids forms hyperbolic knots.
The paper extends knot theory to twisted virtual braids and links.
problem Generalizing knot theory to include twists.
method Introduced twisted virtual braids and proved theorems for twisted links.
result The Alexander and Markov theorems were extended to twisted links.
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. Knot Floer homology stabilizes with twists.
problem Understanding knot behavior through homology.
method Analyzing m full twists on n parallel strands. result Knot Floer homologies stabilize as twists increase.
Study proves bridge and braid indices match for twist positive knots.
problem Determining when bridge and braid indices are equal for knots.
method Used knot Floer torsion order to prove for all twist positive knots.
result Bridge and braid indices coincide for all twist positive knots.
Proved colored HOMFLY-PT polynomials for specific knots.
problem Calculating colored HOMFLY-PT polynomials for specific knots.
method Rigorous mathematical proof for trefoil, figure-eight, and twist knots.
result Colored HOMFLY-PT polynomials expressed as sums for different knots.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister t…
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
The study connects twist positivity to L-space knots and concordance.
problem Understanding the relationship between twist positivity and knot concordance.
method Analyzing Alexander polynomials and braid indices of knots.
result For twist positive L-space knots, the braid index equals the bridge index.
The paper studies knots formed by twisting a circle around a base knot and conjectures a linear growth in crossing numbers.
problem Understanding the growth rate of crossing numbers in twist families of knots.
method Introduced the stable crossing number and used geometric wrapping and algebraic winding to establish conjectures.
result The crossing number of Kn grows like nη(η−1) as no∞ for coherent twist families. 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.
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.
Study on unknotting twisted knots using arc shift and region arc shift moves.
problem Unknotting twisted knots and finding bounds for region arc shift number.
method Introduced arc shift move and region arc shift move for twisted knots.
result Found families of twisted knots with specific arc shift and region arc shift numbers.
The twisted torus knots lie on the standard genus 2 Heegaard surface for S3, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…
New method distinguishes knots and knotted surfaces.
problem Distinguishing knots and knotted surfaces.
method Twisted set-theoretic Yang-Baxter solutions and Alexander numbering.
result Distinguished 2-twist spun trefoil from its reverse. Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
New 2-knots found with same knot group but different quandles.
problem Identifying 2-knots with identical knot groups but distinct quandles.
method Analyzing knot quandles of twist spins.
result First example of 2-knots with same knot group but different quandles.
Paper calculates the ribbonlength of twisted torus knots.
problem Determining the ribbonlength of twisted torus knots.
method Analyzes the ribbonlength of twisted torus knots Tp,q;r,s and provides upper bounds. result Ribbonlength of Tp,q;r,s is bounded by 2(max{p,q,r}+∣s∣r) and 2(p+(∣s∣−1)r) under specific conditions. Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist knots.
This paper extends danceability concept to twisted virtual knots.
problem Danceability of knot diagrams on non-orientable surfaces.
method Expanding danceability concept to twisted virtual knot diagrams.
result Danceability properties of twisted virtual knots.
Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.
problem Computing Jones polynomial for a specific family of links.
method Computed Jones polynomial for T((p,q),(2,s)) links, focusing on s=2n. result Jones polynomial is trivial if and only if the knot is trivial.
We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than…
The twisting number of a ribbon knot is at least as large as its doubly slice genus.
problem Proving a lower bound for the twisting number of ribbon knots in terms of their doubly slice genus.
method Analyzing symmetric unions and tangle replacements to establish the bound.
result Ribbon knots have arbitrarily high twisting numbers, matching their doubly slice genus.
Characterizes groups of branched twist-spun knots.
problem Understanding the groups of branched twist-spun knots.
method Characterization through 3-manifold groups and conjectural algebraic approach.
result Each group is the group of at most finitely many branched twist spins.
Proves bounds on ribbonlength for various knot types.
problem Finding bounds on the infimal folded ribbonlength of knot types.
method Applied techniques to multi-twist Möbius bands and knot constructions.
result Established bounds for (2,q) torus knots and twist knots. Study calculates twisted Alexander polynomials for Montesinos knots.
problem Tackles the calculation of twisted Alexander polynomials for Montesinos knots.
method Uses SL2(C)-representations to calculate leading coefficients and degrees of the polynomials. result Obtains non-monic twisted Alexander polynomials for some nonfibered knots.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.
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.
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.
Computes Jones polynomial for specific knots.
problem Computing Jones polynomial for double twist knots.
method Using cyclotomic expansion and Kauffman bracket skein theory.
result Answers a question about Jones polynomial for specific knots.
The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, K and K′, intersecting at two points transversely. Each of K and K′ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along…
We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation C(2n,3). We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.
Paper discusses groups where twisted Alexander polynomials vanish.
problem Understanding groups with vanishing twisted Alexander polynomials.
method Introduced and studied twisted Alexander vanishing groups, constructed knots, and analyzed representations.
result Every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot 52, which is the (−2)-twist knot, by Naylor and Rolfsen.
Simplified A-polynomial calculation for twisted knots.
problem Calculating A-polynomials for twisted knots.
method Observation of exchange relations in cluster algebra and proof of Laurent phenomenon.
result Simplified method for calculating A-polynomials for twisted knots.
New satellite knots found that can't be represented by positive braids with full twists.
problem Satellite knots that cannot be represented by positive braids with full twists.
method Analyzing positive minimal braids and their closures to find satellite knots.
result Infinitely many satellite knots with Lorenz patterns and companions are not Lorenz knots.
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.
We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa an…
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.