Pretzel Knots Research Articles

BANDED SURFACES, BANDED LINKS, BAND INDICES AND GENERA OF LINKS

Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of bands required to present the link as boundaries of an unknotted (flat) banded surface. Banded links of small (flat, respectively) band index are considered here. Some upper bounds are provided for these band indices of a link using braid representatives and canonical Seifert surfaces of the link. The relation between the band indices and genera of links is studied and the band indices of pretzel knots are calculated.

A GOOD PRESENTATION OF (-2, 3, 2s + 1)-TYPE PRETZEL KNOT GROUP AND ℝ-COVERED FOLIATION

Let Ksbe a (-2, 3, 2s + 1)-type pretzel knot (s ≧ 3) and EKs(p/q) be a closed manifold obtained by Dehn surgery along Kswith a slope p/q. We prove that if q > 0, p/q ≧ 4s + 7 and p is odd, then EKs(p/q) cannot contain an ℝ-covered foliation. This result is an extended theorem of a part of works of Jun for (-2, 3, 7)-pretzel knot.

ON JONES POLYNOMIALS OF ALTERNATING PRETZEL KNOTS

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

The Noncommutative A-Polynomial of (−2, 3, n) Pretzel Knots

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 (noncommutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the Kp =(−2, 3, 3+2p) pretzel knots for p=−5, … , 5. This is a particularly interesting family, since the pairs (Kp , −K −p ) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the noncommutative A-polynomial complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, supports the AJ conjecture for knots with reducible A-polynomial, and numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the volume conjecture for the above-mentioned pretzel knots.

The canonical genus for Whitehead doubles of a family of alternating knots

For any given integer r⩾1 and a quasitoric braid βr=(σr−ϵσr−1ϵ⋯σ1(−1)rϵ)3 with ϵ=±1, we prove that the maximum degree in z of the HOMFLYPT polynomial PW2(βˆr)(v,z) of the doubled link W2(βˆr) of the closure βˆr is equal to 6r−1. As an application, we give a family K3 of alternating knots, including (2,n)-torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in K3 coincides with the canonical genus of its Whitehead double. Consequently, we give a new family K3 of alternating knots for which Trippʼs conjecture holds.

A DETERMINANT FORMULA FOR THE JONES POLYNOMIAL OF PRETZEL KNOTS

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above. Evaluations of these activity words are related to the chain complex for the Champanerkar–Kofman spanning tree model of reduced Khovanov homology.

THE KHOVANOV HOMOLOGY OF (p, -p, q) PRETZEL KNOTS

In this paper, we compute the Khovanov homology over ℚ for (p, -p, q) pretzel knots for 3 ≤ p ≤ 15, p odd , and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p, -p, q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over ℚ or ℤ). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.

THE REPRESENTATIVITY OF PRETZEL KNOTS

In the previous paper, the representativity of a non-trivial knot K was defined as [Formula: see text] where [Formula: see text] is the set of all closed surfaces in S3 containing K and [Formula: see text] is the set of all compressing disks for F in S3. Then an inequality r(K) ≤ b(K) has shown, where b(K) denotes the bridge number of K. This shows that a (p, q, r)-pretzel knot has the representativity less than or equal to 3. In this paper, we will show that a (p, q, r)-pretzel knot has the representativity 3 if and only if (p, q, r) is either ±(-2, 3, 3) or ±(-2, 3, 5). We also show that an algebraic knot has the representativity less than or equal to 3.

Cosmetic crossings and Seifert matrices

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.

Pretzel knot compared with standard suture knots

Aim of this paper is to compare the biomechanical properties of the Pretzel knot with the selected arthroscopic knots and to test the mechanical safety of this knot. The hypothesis was that the Pretzel knot may have equal or better mechanical properties than the arthroscopic knots tested in this study. SMC, Giant, Dines, Nicky's, Tennessee Slider knots were chosen for comparison. Original and backed configuration of knots were prepared with no. 2 braided polyester sutures. Cyclic loading and load to failure tests were performed. Parameters tested were loop security, maximum elongation and load at failure. Number of steps needed for preparation of each knot was also recorded. One-way ANOVA for repeated measures was performed for all knots. Steps needed for preparation were 2 for Pretzel and Tennessee Slider, 3 for Nicky's, SMC and Dines and 4 for Giant. The Pretzel knot in original configuration had significantly better loop security compared to SMC, Dines, Nicky's and Tennessee knots (P=0.01). Loads at failure were not significantly different between knots in their original configurations. Giant and Dines knots showed better maximum elongations in cyclic loading (P=0.0059) and load to failure tests (P=0.0001). Backing up the knots with 3-RHAPs eliminated most of these significant differences. Clinically, simplicity and safety are two important components in arthroscopic knot tying. Despite its simple configuration, Pretzel knot has some similar and increased mechanical properties compared to the tested knots that are widely used.

On the geometry of the slice of trace-free $${{SL_2(\mathbb{C})}}$$ -characters of a knot group

Let K be a knot in an integral homology 3-sphere Σ with exterior EK, and let B2 denote the two-fold branched cover of Σ branched along K. We construct a map Φ from the slice of trace-free \({{{\rm SL}_2(\mathbb{C})}}\) -characters of π1(EK) to the \({{{\rm SL}_2(\mathbb{C})}}\)-character variety of π1(B2). When this map is surjective, it describes the slice as the two-fold branched cover over the \({{{\rm SL}_2(\mathbb{C})}}\)-character variety of B2 with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each metabelian character can be represented as the character of a binary dihedral representation of π1(EK). The map Φ is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.

Representation spaces of pretzel knots

We study the representation spaces $R(K;\bf{i})$ 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 binary dihedral.

Exceptional surgeries on [formula omitted]-pretzel knots

We give a complete description of exceptional surgeries on pretzel knots of type ( − 2 , p , p ) with p ⩾ 5 . It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a twisted I-bundle over the Klein bottle. We show that the other is homeomorphic to the one obtained by certain Dehn filling on the magic manifold. On the other hand, we show that all such pretzel knots admit no Seifert fibered surgeries.

The slice-ribbon conjecture for 3-stranded pretzel knots

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-Gordon.

The Jones slopes of a knot

The paper introduces the slope conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2,3,n) pretzel knots.

Rational Witt classes of pretzel knots

In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. We also obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference Fifty years since Milnor and Fox held at Brandeis University on June 2-5, 2008.

KHOVANOV HOMOLOGY AND RASMUSSEN'S s-INVARIANTS FOR PRETZEL KNOTS

We calculate the rational Khovanov homology of a class of pretzel knots by using the spectral sequence constructed by Turner. Moreover, we determine Rasmussen's s-invariant of almost all pretzel knots P(p, q, r) by using Turner's spectral sequence, a sharper slice-Bennequin inequality, and a skein inequality.

COMPLEXITY, HEEGAARD DIAGRAMS AND GENERALIZED DUNWOODY MANIFOLDS

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

THE DISJOINT CURVE PROPERTY AND BRIDGE SURFACES

We show that every bridge surface of certain types of (1, 1) prime knot has the disjoint curve property. Also we determine when a bridge surface of a pretzel knot of type (.2, 3, n) has the disjoint curve property.

Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Given a knot complement X and its p-fold cyclic cover X p → X, we identify twisted polynomials associated to $${GL_1\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X p ) with twisted polynomials associated to related $${GL_p\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3, 7, 9, 11, 15), corresponding to permutations of (7, 9, 11, 15), represent distinct concordance classes.