Pretzel Links Research Articles

2-Adjacency between some special links and pretzel links

This paper studies 2-adjacency between a 3-strand pretzel link and one of the Hopf link, the Solomon’s link and the Whitehead link by using the results that have been obtained about 2-adjacency between knots or links and their polynomials and etc. This paper shows that of all 3-strand pretzel links, only ordinary pretzel links are 2-adjacent to the Hopf link or the Solomon’s link or the Whitehead link. Conversely, these special links are not 2-adjacent to any other 3-strand pretzel links, except for themselves, respectively.

Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

Trace-free SL(2, ℂ)-representations of Montesinos links

Given a link [Formula: see text], a representation [Formula: see text] is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when [Formula: see text] is a Montesinos link.

The classification of quasi-alternating Montesinos links

In this paper, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link L L is quasi-alternating if and only if its double branched cover is an L-space, and bounds both a positive definite and a negative definite 4 4 -manifold with vanishing first homology.

Hoste’s Conjecture and Roots of Link Polynomials

In relation to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots, we prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies $${| z^{1/2}-z^{-1/2} | < 2}$$ . We extend our result to properties of zeros for some Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links.

Bridge spectra of cables of 2-bridge knots

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Montesinos knots, Hopf plumbings, and L-space surgeries

Using Hirasawa–Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman–Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on $S^3$. We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.

A note on graphs and rational balls

In this short note we study some particular graphs associated to small Seifert spaces and Montesinos links. The study of these graphs leads to new examples of Seifert manifolds bounding rational homology balls and Montesinos links bounding smoothly and properly embedded surfaces (possibly not orientable) in the 4 ball with Euler characteristic equal to 1.

Link homology and equivariant gauge theory

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links.

Mutation invariance of the arc index for some montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

Involutory quandles of (2,2,r)-Montesinos links

In this paper, we show that Montesinos links of the form [Formula: see text], which we call [Formula: see text]-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the [Formula: see text]-pretzel links. We also describe some properties of these quandles.

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.

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.

Volumes of Montesinos links

We show that the volume of any Montesinos link can be bounded above and below in terms of the combinatorics of its diagram. This was known for Montesinos links with at most two tangles, and those with at least five tangles. We complete the result for the remaining cases.

Character varieties of (−2,2m + 1,2n)-pretzel links and twisted Whitehead links

We consider some families of links which include [Formula: see text]-pretzel links and twisted Whitehead links. We calculate the [Formula: see text]-character varieties of these families, and determine the number of irreducible components of these character varieties.

Isometry Groups of Some Dunwoody Manifolds

Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly 𝕊3) branched over (1,1)-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.

A remark on 2-adjacency relation between Montesinos knots and 2-bridge knots

We show that for every 2-bridge knot K, there is an infinite family of Montesinos knots {Kq} such that each Kq is 2-adjacent to K. We also give a new example of 2-adjacency relation between 2-bridge knots.

A note on quasi-alternating Montesinos links

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.

Colored HOMFLY polynomials for the pretzel knots and links

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 structure: are decomposed into a sum of a product of g + 1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU q (N ) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

RASMUSSEN INVARIANTS OF SOME 4-STRAND PRETZEL KNOTS

Abstract. It is known that there is an in nite family of generalpretzel knots, each of which has Rasmussen s-invariant equal to thenegative value of its signature invariant. For an instance, homo-logically ˙-thin knots have this property. In contrast, we nd anin nite family of 4-strand pretzel knots whose Rasmussen invariantsare not equal to the negative values of signature invariants. 1. IntroductionKhovanov [7] introduced a graded homology theory for oriented knotsand links, categorifying Jones polynomials. Lee [10] de ned a variant ofKhovanov homology and showed the existence of a spectral sequence ofrational Khovanov homology converging to her rational homology. Leealso proved that her rational homology of a knot is of dimension two.Rasmussen [13] used Lee homology to de ne a knot invariant sthat isinvariant under knot concordance and additive with respect to connectedsum. He showed that s(K) = ˙(K) if Kis an alternating knot, where˙(K) denotes the signature of K.Suzuki [14] computed Rasmussen invariants of most of 3-strand pret-zel knots. Manion [11] computed rational Khovanov homologies of allnon quasi-alternating 3-strand pretzel knots and links and found theRasmussen invariants of all 3-strand pretzel knots and links.For general pretzel knots and links, Jabuka [5] found formulas fortheir signatures. Since Khovanov homologically ˙-thin knots have sequal to ˙, Jabuka’s result gives formulas for sinvariant of any quasi-alternating pretzel knot. Note that quasi-alternating pretzel knots are