arXiv research

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

Exceptional Dehn surgeries have been classified for 2-bridge knots and Montesinos knots of length at least 4. In this paper we classify all toroidal Dehn surgeries on Montesinos knots of length 3.

Study calculates twisted Alexander polynomials for Montesinos knots.

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 …

New method for simplifying knots with specific properties.

We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.

Study counts specific surfaces in Montesinos knots with 4 rational tangles.

We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups.

We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.

The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions.

This note corrects errors in Hatcher and Oertel's table of boundary slopes of Montesinos knots which have projections with 10 or fewer crossings.

Method determines SL(2,C) character variety for Montesinos knots.

We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.

Study shows torsion in knot module for specific Montesinos knots.

It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.

We give (1) an upper bound on the denominators of numerical boundary slopes and (2) an upper bound on the differences between two numerical boundary slopes, for Montesinos knot exteriors.

The paper classifies knot Floer complexes of low width, simplifying knot bases.

It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.

Proves a property of certain 3D knots and links.

We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.

A knot's thickness is measured by its β invariant, a new numerical invariant.

Study shows not all ribbon knots can be symmetric unions.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that…

The Slope Conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of incompressible surfaces. Our aim is to prove the Slope Conjecture for Montesinos knots, and to match parameters of a state-formula for the colored Jones polynomial of such knots with the parameters that describe thei…

In this paper, two lower bounds on the diameters of the boundary slope sets are given for Montesinos knots. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

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…

The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.

Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra …

The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots $M(\frac{1}{r},\frac{1}{s-\frac{1}{u}},\frac{1}{t} )$ with $r,u,t$ odd, $s$ even and $u\leq-1$, $r<-1<1<s,t$.

We show that there are Montesinos knots with $n+1\geq 4$ tangles whose character varieties contain arbitrarily many irreducible components of dimension $d$ for any $1\le d\le n-2$. Moreover, these irreducible components can be chosen so that the trace of the meridian is non-constant.

Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many…

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …

Character varieties of prime knots have high-dimensional components.

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…

The paper explores knots with equal bridge and braid index, conjecturing they have a unique equilibrium state.

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

A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.

C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. Afte…

This paper gives a complete classification of all alternating knots with tunnel number one, and all their unknotting tunnels. We prove that the only such knots are two-bridge knots and certain Montesinos knots.

We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.

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.

We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.

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 param…

Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.

Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel number of all 1655 alternating 11 and 12 crossing knots and of 881 non-alternating 11 and 12 crossing knots. We also find all 5525 Montesinos knots with 14 crossings or fewer.

Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.

The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos k…

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way…