Pretzel Knots Research Articles

Purely cosmetic surgeries and pretzel knots

We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a pretzel knot provide different oriented three-manifolds.

Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

Classical pretzel knots and left orderability

We consider the classical pretzel knots [Formula: see text], where [Formula: see text] are positive odd integers. By using continuous paths of elliptic [Formula: see text]-representations, we show that (i) the 3-manifold obtained by [Formula: see text]-surgery on [Formula: see text] has left orderable fundamental group if [Formula: see text] and (ii) the [Formula: see text]-cyclic branched cover of [Formula: see text] has left orderable fundamental group if [Formula: see text].

Cancellations in the Degree of the Colored Jones Polynomial

We give an alternate expansion of the colored Jones polynomial of a pretzel link which recovers the degree formula in Garoufalidis et al. (Internat. J. Math.31(7):66, 2020). As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle pretzel knots.

Hyperbolic torsion polynomials of pretzel knots

Abstract In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

Representations of the (-2,3,7)-pretzel knot and orderability of Dehn surgeries

We construct a 1-parameter family of SL2(R) representations of the pretzel knot P(−2,3,7). As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.

Crossing number of an alternating knot and canonical genus of its whitehead double

A conjecture proposed by J. Tripp in 2002 and modified by T. Nakamura in 2006 states that the crossing number of any alternating knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including (2,m) torus knots, 2-bridge knots, algebraic alternating knots, alternating pretzel knots and so on. In this paper, we prove that the conjecture is not true for any alternating 3-braid knot which is the connected sum of two torus knots of type (2,m) and (2,n) with odd integers m,n≥3. This results in a new modified conjecture that the crossing number of any prime alternating knot coincides with the canonical genus of its Whitehead double. We prove that this modified conjecture holds for all prime alternating 3-braid knots in addition to the known classes above.

Braid group and leveling of a knot

Any knot [Formula: see text] in genus-1 1-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1,1)-positions of all 2-bridge knots providing upper bounds for their level numbers and also show that the (-2, 3, 7)-pretzel knot has level number two.

Doubly slice odd pretzel knots

We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots.

Sliceness of alternating pretzel knots and links

We show that every alternating pretzel knot satisfies the slice-ribbon conjecture and that every alternating pretzel link with m components (m≥2) is not slice in the strong sense.

Taut foliations, positive 3‐braids, and the L‐space conjecture

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot $P(-2,3,7)$, and indeed along every pretzel knot $P(-2,3,q)$, for $q$ a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 1-bridge braid in $S^3$, where $r <g(K)$.

On spectral sequences from Khovanov homology

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsv\'ath-Szab\'o to the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4,5) torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsv\'ath-Szab\'o spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv\'ath-Szab\'o spectral sequences necessarily lower the delta grading for all pretzel knots.

Smoothly embedding Seifert fibered spaces in 𝑆⁴

Using an obstruction based on Donaldson’s theorem, we derive strong restrictions on when a Seifert fibered space Y = F ( e ; p 1 q 1 , … , p k q k ) Y = F(e; \frac {p_1}{q_1}, \ldots , \frac {p_k}{q_k}) over an orientable base surface F F can smoothly embed in S 4 S^4 . This allows us to classify precisely when Y Y smoothly embeds provided e &gt; k / 2 e &gt; k/2 , where e e is the normalized central weight and k k is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant μ ¯ \overline {\mu } , we make some conjectures concerning Seifert fibered spaces which embed in S 4 S^4 . Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.

Stability and triviality of the transverse invariant from Khovanov homology

We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are naturally transverse links in the standard contact 3-sphere. For any 3-braid β, we show that the transverse invariant of its closure does not vanish whenever the fractional Dehn twist coefficient of β is strictly greater than one. We show that Plamenevskaya's transverse invariant is stable under adding full twists on n or fewer strands to any n-braid, and use this to detect families of braids that are not quasipositive. Motivated by the question of understanding the relationship between the smooth isotopy class of a knot and its transverse isotopy class, we also exhibit an infinite family of pretzel knots for which the transverse invariant vanishes for every transverse representative, and conclude that these knots are not quasipositive.

Twisted Alexander polynomials of $(−2, 3, 2n+1)$-pretzel knots

We calculate the twisted Alexander polynomials of $(−2, 3, 2n+1)$-pretzel knots associated to the family of their $SL_2(\mathbb C)$-representations which contains their holonomy representations.

A Biomechanical and Ease of Learning Comparison Study of Arthroscopic Sliding Knots.

This study aims to compare the biomechanical properties and ease of learning and tying of our novel knot (UM Knot) with other commonly used arthroscopic sliding knots. The Duncan, HU, SMC, Pretzel, Nicky's and square knots were selected for comparisons with UM knot. All knots were prepared with size 2 HiFi® suture by a single experienced surgeon and tested with cyclic loading and load to failure tests. The ease of learning was assessed objectively by recording the time to learn the first correct knot and the total number of knots completed in 5min by surgeons and trainees. The UM knot average failure load is significantly superior to the HU knot (p < 0.05) and comparable to Duncan, SMC, Pretzel and Nicky's knots. According to the ease of learning assessment, UM, Duncan, SMC, Pretzel and Nicky's knots took statistically less time to learn than the HU knot. Although not significant, the failure count due to slippage is fewer in UM knot compared with other knots. This study showed that UM knot is among the easiest knot to learn and tie, along with Duncan, SMC, Pretzel and Nicky's knots. Their biomechanical properties are comparable and their loads to failure were superior to the HU knot.

Character varieties of even classical pretzel knots

Abstract For each even classical pretzel knot P(2k1 + 1, 2k2 + 1, 2k3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters n_0, dots , n_g. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at Tne -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and lambda = q^2 T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and -q^2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” lambda , namely, they are equal to lambda ^2, dots , lambda ^g. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when lambda is pure phase the contributions of lambda ^2, dots , lambda ^g oscillate “faster” than the one of lambda . Hence, we call this type of evolution “nimble”.

2-Adjacency between some special knots and pretzel knot

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

On the 2-head of the colored Jones polynomial for pretzel knots

Abstract In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.