Factorization of DE coefficients is violated in antiparallel triple pretzels, but described elegantly.
problem Understanding the origins of factorization in double braids and its extension to antiparallel triple pretzels.
method Defect-preserving deformation from trefoil to antiparallel triple pretzels, analysis of DE coefficients.
result Factorization of DE coefficients is violated but described by an elegant formula for symmetric representations.
With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new struc…
Steinhaus conjectured that every closed oriented C1-curve has a pair of anti-parallel tangents. Porter disproved the conjecture by showing that there exist curves with no anti-parallel tangents. Colin Adams rised the question of whether there exists a nontrivial knot in R3 which has no parallel or antiparallel t…
Defect of knot polynomials remains invariant under certain braid substitutions.
problem Invariance of knot polynomial defects under specific transformations.
method Investigation of defect invariants under antiparallel and parallel braid substitutions.
result Defect remains unchanged under antiparallel braid substitutions and changes by half the added length under parallel braid substitutions.
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…
In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …
A method to convert pretzel links into braids.
problem Representing pretzel links as braids.
method General procedure to convert any pretzel link into a braid.
result A specific braid word corresponding to the pretzel link.
This study limits the number of pretzel links with a specific Jones polynomial span.
problem Determining the number of pretzel links with a given Jones polynomial span.
method Developed an algorithm to decide if a knot is pretzel and used it to identify all pretzel knots up to nine crossings.
result Identified all pretzel knots up to nine crossings, proving 812 is not pretzel. Cosmetic surgeries on pretzel knots are unique.
problem Understanding unique three-manifolds from surgeries on knots.
method Analyzing pretzel knots through Dehn surgeries.
result All pretzel knots satisfy the cosmetic surgery conjecture.
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…
Classifies fibered ribbon pretzels, except for a few cases.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
Study completes braid index determination for all pretzel links.
problem Determine braid indices of all pretzel links.
method Divided pretzel links into three types and derived formulas for braid indices.
result Close to determining braid indices for all Type 3 pretzel links.
Explicit formulas for pretzel knots' Alexander polynomials.
problem Alexander polynomial of pretzel knots
method Provided explicit formulas
result Characterization of pretzel knots with trivial Alexander polynomial
We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three.
The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, P(c), for any given crossing number c. result The number of alternating pretzel links grows exponentially with the crossing number.
Computed involutive knot invariants for specific pretzel knots.
problem Computing involutive knot invariants for a specific class of knots.
method Computed involutive knot invariants for pretzel knots of the form P(-2,m,n) with m and n odd and ≥ 3.
result Computed involutive invariants for a specific class of knots.
A necessary and sufficient condition for an oriented pretzel surface to be quasipositive yields an estimate for the slice genus of the boundary of an arbitrary oriented pretzel surface.
Paper generalizes pretzel links using spatial graphs.
problem Classical pretzel links need a generalization.
method Introduces graph-pretzel links based on spatial graph projections.
result Constructs an infinite family of distinct ribbon knots.
The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.
problem Classifying pretzel links based on their self delta-equivalence.
method Using Conway polynomials to determine self delta-equivalence for links with 2 or more components.
result Necessary and sufficient conditions for self delta-equivalence of pretzel links with 3 or more components.
This paper determines the braid indices for non-alternating pretzel links.
problem Determining the braid index of non-alternating pretzel links.
method Classified pretzel links into three types and computed braid indices for Type 1 and Type 2.
result Precise braid indices for all non-alternating Type 1 and Type 2 pretzel links.
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology…
A pretzel knot K is called odd if all its twist parameters are odd, and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5-stranded pretzel knots are mutant ribbon. We d…
Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.
problem Characterizing chirally cosmetic surgeries on specific knot types.
method Recent methods of Ichihara, Ito, and Saito applied to genus 2 and 3 alternating odd pretzel knots.
result Most genus 2 and 3 alternating odd pretzel knots do not admit chirally cosmetic surgeries.
The study classifies χ−slice pretzel links and Seifert fiber spaces.
problem Understanding χ−slice pretzel links and their properties. method Analyzing the sliceness of pretzel knots and extending results to pretzel links.
result Complete classifications of positive and negative pretzel links that are χ−slice, and partial classifications of 3-stranded and 4-stranded pretzel links. Short note on braid index and quasipositivity of certain pretzel knots.
problem Calculating braid index and identifying quasipositive status for specific pretzel knots.
method Used Morton-Franks-Williams inequalities and Khovanov-Rozansky concordance homomorphisms.
result Determined braid index and identified quasipositivity for knots with even crossings in one strand.
We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon or has trivial Alexander polynomial. (By work of [FS85], a nontrivial odd 3-strand pretzel knot K cannot both be ribbon…
The paper calculates the Δ-unknotting number for positive pretzel knots.
problem Determining the minimum number of Δ-moves to simplify pretzel knots.
method Analyzing the Conway polynomial and computing Δ-unknotting numbers.
result The Δ-unknotting number for positive pretzel knots is maximized by a specific knot type.
A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.
We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.
The paper constructs paths of SL2(R) representations for pretzel knots and shows left-orderability conditions.
problem Left-orderability of knot surgery manifolds.
method Explicit construction of continuous paths of SL2(R) representations.
result Fundamental groups of certain knot surgeries are left-orderable.
We prove that an odd pretzel knot is doubly slice if it has 2n+1 twist parameters consisting of n+1 copies of a and n copies of −a for some odd integer a. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots.
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
We study the representation spaces R(K;i) as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots P(p,q,r) with p,q,r pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not b…
New knot groups found to be bi-orderable using pretzel knots.
problem Understanding bi-orderability of knot groups.
method Applied Mayland's technique to pretzel knots to show their commutator subgroups are residually-torsion-free nilpotent.
result Found new examples of bi-orderable knot groups for non-fibered or alternating knots.
Researchers study rational and pretzel knots using affine group representations.
problem Understanding the structure and properties of rational and pretzel knots.
method Constructing representations of knot groups into the affine group AGL(1,ℂ) via a TQFT valued in spans of singular vector bundles.
result Closed-form expressions for Alexander polynomials and bounds on their zeros for rational and pretzel knots.
Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…
We compute the reduced Khovanov homology of 3-stranded pretzel links. The coefficients are the integers with the "even" sign assignment. In particular, we show that the only homologically thin, non-quasi-alternating 3-stranded pretzels are P(-p,p,r) with p an odd integer and r greater than or equal to p (these were sho…
We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we obtain this polynomial explicitly for the K_p = (-2, 3, 3+2p) pretzel knots for p = -…
We calculate the universal character ring of a class of two-generator, one-relator groups. As an application we give a less technical proof of a result in [LT] on the universal character ring of the (-2,3,2n+1)-pretzel knot. We also give an elementary proof of a result in [Ma] on the character variety of the (-2,3,2n+1…
Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.
problem Calculating colored Jones polynomials for general oriented links is difficult.
method Using Kuperberg's linear skein theory, they focus on one-row polynomials for pretzel links.
result Existence of tails for specific pretzel knots' Jones polynomials is shown.
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…
We explicitly calculate the universal character ring of the (-2,2m+1,2n)-pretzel link and show that it is reduced for all integers m and n.
If p/q > 18, p is odd, and p/q=37/2, (p,q)-Dehn surgery for the (-2,3,7)-pretzel knot produces a 3-manifold without Reebless foliation.
Study the JSJ-decomposition of a specific 3-manifold.
problem Classify the JSJ-decomposition of a 3-manifold from 0-surgery on a pretzel knot.
method Utilize the classification of exceptional fillings of minimally twisted five-chain links.
result Determine the JSJ-decomposition of the 3-manifold.
Study provides concrete examples of knot slopes.
problem Finding explicit characterizing slopes for knots.
method Concrete examples for the (-2,3,7)-pretzel knot.
result Explicit characterizing slopes for the knot 12n242. Study proves non-left-orderability of 3-manifolds derived from specific knots.
problem Proving non-left-orderability of knot groups in 3-manifolds.
method Analyzing fundamental groups of cyclic branched covers of pretzel knots.
result Non-left-orderability of fundamental groups of n-fold cyclic branched covers of P(3,−3,−2k−1) for all integers k and n≥1.