Pretzel Knots Research Articles

A Calculation of the Hyperbolic Torsion Polynomial of a Pretzel Knot

In this short note, we calculate the highest degree term of the hyperbolic torsion polynomial of a pretzel knot with three tangles. It gives a supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot.

Left-orderablity for surgeries on (−2,3,2s + 1)-pretzel knots

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a (−2,3,2s+1)-pretzel knot (s≥3) with slope pq is not left orderable if pq≥2s+3, and that it is left orderable if pq is in a neighborhood of zero depending on s.

Distance to longitude

Let [Formula: see text] be a hyperbolic knot in the [Formula: see text]-sphere. If a [Formula: see text]-Dehn surgery on [Formula: see text] produces manifold with an embedded Klein bottle or essential [Formula: see text]-torus, then we prove that [Formula: see text], where [Formula: see text] is the genus of [Formula: see text]. We obtain different upper bounds according to the production of a Klein bottle, a non-separating [Formula: see text]-torus, or an essential and separating [Formula: see text]-torus. The well known examples which are the figure eight knot and the pretzel knot [Formula: see text] reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed [Formula: see text]-manifolds.

Alternating knots with Alexander polynomials having unexpected zeros

We give concrete counterexamples to a conjecture due to J. Hoste. Namely, we present alternating knots whose Alexander polynomials have zeros with real parts less than −1. Our knots are pretzel knots having alternating Montesinos diagrams.

Character varieties of odd classical pretzel knots

We determine the [Formula: see text]-character variety for each odd classical pretzel knot [Formula: see text], and present a method for computing its A-polynomial.

WEIGHT ELEMENTS OF THE KNOT GROUPS OF SOME THREE-STRAND PRETZEL KNOTS

A knot group has weight one, so is normally generated by a single element called a weight element of the knot group. A meridian is a typical weight element, but some knot groups admit other weight elements. We show that for some infinite classes of three-strand pretzel knots and all prime knots with up to eight crossings, the knot groups admit weight elements that are not automorphic images of meridians.

Finite Dehn surgeries on knots in S3

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For $D$-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that $4m$ and $4m+4$ are characterizing slopes for the torus knot $T(2m+1,2)$ for each $m\geq 1$.

Affine cubic surfaces and character varieties of knots

It is known that the fundamental group homomorphism π1(T2)→π1(S3∖K) induced by the inclusion of the boundary torus into the complement of a knot K in S3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL2-character varieties of the corresponding fundamental groups. In our earlier work [3], we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C3. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings.

Slice implies mutant ribbon for odd 5–stranded pretzel knots

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 do this in stages by first showing that 5-stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. Next, we show that any odd, 5-stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.

The topological sliceness of 3–strand pretzel knots

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 and have $\Delta_K(t)=1$.) We also show that topologically slice even 3-strand pretzel knots (except perhaps for members of Lecuona's exceptional family of [Lec13]) must be ribbon. These results follow from computations of the Casson-Gordon 3-manifold signature invariants associated to the double branched covers of these knots.

Infinitely many prime knots with the same Alexander invariants

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike, other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels.

Pretzel Knots and $q$-Series

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 this paper, we study $q$-series identities coming from the colored Jones polynomial of pretzel knots. We prove a false theta function identity that goes back to Ramanujan and we give a natural generalization of this identity using the tail of the colored Jones polynomial of Pretzel knots. Furthermore, we compute the tail for an infinite family of Pretzel knots and relate it to false theta function-type identities.

Distinguishing mutant pretzel knots in concordance

We prove that many four-strand pretzel knots of the form [Formula: see text] are not topologically slice, even though their positive mutants [Formula: see text] are ribbon. We use the sliceness obstruction of Kirk and Livingston [Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635–661], related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

A class of knots with simple SU(2)-representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of rank equal to $\det(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.

On the AJ conjecture for cable knots

We study the AJ conjecture for $(r,2)$-cables of a knot, where $r$ is an odd integer. Using skein theory, we show that the AJ conjecture holds true for most $(r,2)$-cables of some classes of two-bridge knots and pretzel knots.

2-Adjacency between pretzel links and the trivial link

We study 2-adjacency between a 3-strand pretzel link with two components and the trivial link, and whether the trivial knot is 2-adjacent to a pretzel knot. By investigating mainly the coefficients of their Conway polynomials, Jones polynomials and etc., we show that any nontrivial pretzel link with two-components is not 2-adjacent to the trivial link, and vice versa. Moreover, we also show that the trivial knot is not 2-adjacent to any nontrivial pretzel knot.

Cosmetic surgery in L-spaces and nugatory crossings

The cosmetic crossing conjecture (also known as the “nugatory crossing conjecture”) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.

Vassiliev invariants for pretzel knots

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on [Formula: see text] parameters [Formula: see text]. These invariants are symmetric polynomials in [Formula: see text] whose degree coincide with their order. We also discuss their topological and integer-valued properties.

On the Combinatorics of Smoothing

Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of the spectral theory of graphs to approach such problems systematically. We give an application to counting subdiagrams of pretzel knots which have one component after oriented and unoriented smoothings.

Pretzel knots with L-space surgeries

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.