Characterizes topological sliceness of 3-strand pretzel knots.
problem Determining the topological sliceness of 3-strand pretzel knots.
method Computations of Casson-Gordon 3-manifold signature invariants for double branched covers.
result Odd 3-strand pretzel knots are topologically slice if and only if they are ribbon or have trivial Alexander polynomial.
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…
Study shows most knots in a family are not slice.
problem Determining which 3-stranded pretzel knots are slice.
method Analyzing a specific infinite family of knots and proving their non-slice properties.
result Four-fifths of the remaining knots in the family are not slice.
We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Go…
Study corrects previous work on knot Floer homology of certain pretzel knots.
problem Computing the Knot Floer Homology of specific pretzel knots.
method Applied peculiar modules theory for Floer homology of 4-ended tangles, using immersed curve interpretation.
result Corrected the rank of Knot Floer Homology for some pretzel knots.
We provide a partial classification of the 3-strand pretzel knots K=P(p,q,r) with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with r even. We discover that there are only four possible subfamilies which may satis…
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…
Diagrammatic method calculates knot invariant from tangle decompositions.
problem Computing Rasmussen's invariant for knots.
method Diagrammatic approach using tangles and cobordisms.
result Computed s-invariants of all 3-strand pretzel knots. 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. Study shows how certain knots and tori are detected by ideal points in character varieties.
problem Detecting essential twice-punctured tori in character varieties.
method Extending Tillmann's limiting character method to new families of knots and tori.
result Explicit determination of limiting characters at ideal points for specific knots and tori.
Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's di…
Study on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
Paper calculates Racah matrices for 3-strand knots, validating conjectures.
problem Systematic description of colored knot polynomials.
method Highest weight method with Gelfand-Tseitlin tables.
result Explicit Racah matrices and polynomials for 3-strand knots up to 10 crossings.
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.
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. 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.
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
Proves conditions for odd pretzel knots to be doubly slice.
problem Identifying conditions for odd pretzel knots to be doubly slice.
method Analyzes knots with specific twist parameters and odd integers.
result Identifies all doubly slice odd pretzel knots.
HZ transform applied to knot polynomials reveals hyperbolic knot structures.
problem Understanding the structure of knot polynomials and their factorisability.
method Applying the Harer-Zagier transform to knot polynomials and character expansions.
result Construction of an infinite family of hyperbolic knots and proof of factorisability in the 3-strand case.
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.
New proof for a specific class of knots satisfying a weaker Slice-Ribbon Conjecture.
problem Proving a specific class of knots are mutant ribbon.
method Proved by showing infinite order and non-slice properties of knots.
result All slice, odd, 5-stranded pretzel knots are mutant ribbon.
The paper calculates HOMFLY polynomials for 3-strand knots using a new method.
problem Systematic description of colored knot polynomials for arbitrary representations.
method Efficient highest-weight method to find inclusive Racah matrices.
result Explicitly found HOMFLY polynomials for 3-strand knots and confirmed conjectures.
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.
Study character varieties of odd classical pretzel knots.
problem Determine mSL(2,C)-character varieties for odd classical pretzel knots. method Compute A-polynomial for each knot.
result Character varieties of odd classical pretzel knots computed.
Study counts Kauffman states of pretzel knots using generating polynomials.
problem Counting Kauffman states of pretzel knots.
method Explores generating polynomials for specific pretzel knot classes.
result Collects coefficients in generating polynomial expansions.
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.
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.
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.
Study character varieties of even pretzel knots, determining their mSL(2,C)-representations.
problem Determine character varieties of even classical pretzel knots.
method Compute irreducible mSL(2,C)-representations of even classical pretzel knots. result Clarify the steps to compute the A-polynomial of even classical pretzel knots.
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.
Researchers compute Vassiliev invariants for pretzel knots up to order six.
problem Computing Vassiliev invariants for pretzel knots of arbitrary complexity.
method Used Vassiliev invariants up to order six, parameterized by g+1 pretzel knot parameters. result Found symmetric polynomials in n1,…,ng+1 whose degree matches their order. 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.
The study of pretzel knots' colored Jones polynomials yields new q-series identities.
problem Investigating q-series identities from pretzel knots' colored Jones polynomials.
method Analyzing the tail of the colored Jones polynomial for pretzel knots.
result Proves a false theta function identity and generalizes it.
New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.
problem Calculating colored HOMFLY-PT polynomials for links with arbitrary symmetric representations.
method Using quantum Racah coefficients (6j-symbols) of Uq(sl2) to simplify the evaluation. result Multi-colored link polynomials H[r1],[r2] for a specific link L7a3 are successfully evaluated. Formula for 2-head of colored Jones polynomial for pretzel knots proved.
problem Calculating the 2-head of colored Jones polynomial for pretzel knots.
method Skein-theoretic techniques and stability properties of coefficients.
result Formula for 2-head of colored Jones polynomial proved for pretzel knots.
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…
Proves conjectures for pretzel knots using polynomial degrees.
problem Proving conjectures about pretzel knots.
method Using Hatcher-Oertel algorithm and colored Jones polynomial.
result Maximal degrees of colored Jones polynomial determine boundary slopes.
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.
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. 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…
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.
We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.
Study calculates Alexander polynomials for a specific knot type.
problem Determining knot properties like genus and fiberedness.
method Calculates twisted Alexander polynomials for (−2,3,2n+1)-pretzel knots. result Supports Dunfield, Friedl, and Jackson's conjecture.
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 show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
The paper calculates Racah matrices for up to 3 strands of knots and links.
problem Systematic description of colored knot and link invariants.
method Highest weight method and use of Racah matrices.
result Explicit answers for Racah matrices and colored polynomials for 3-strand knots and links.
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.