arXiv research

Classifies fibered ribbon pretzels, except for a few cases.

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 d…

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…

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…

Paper generalizes pretzel links using spatial graphs.

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-Go…

Study shows most knots in a family are not slice.

Defines slice depth for 2-knots and sets upper bounds for specific knots.

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

This paper finds bounds on the ribbonlength of knots and links with up to 9 crossings.

Kirby diagrams for exotic R^4's constructed from specific knot complements.

Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while …

Study on the ribbonlength of knots and links, improving upper bounds.

A method to convert pretzel links into braids.

This study limits the number of pretzel links with a specific Jones polynomial span.

Cosmetic surgeries on pretzel knots are unique.

We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…

Study completes braid index determination for all pretzel links.

Explicit formulas for pretzel knots' Alexander polynomials.

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.

The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.

Computed involutive knot invariants for specific pretzel knots.

A necessary and sufficient condition for an oriented pretzel surface to be quasipositive yields an estimate for the slice genus of the boundary of an arbitrary oriented pretzel surface.

The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.

This paper determines the braid indices for non-alternating pretzel links.

New formula recovers degree of colored Jones polynomials for pretzel knots.

The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology…

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

Short note on braid index and quasipositivity of certain pretzel knots.

The paper calculates the Δ-unknotting number for positive pretzel knots.

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.

This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.

We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.

The paper constructs paths of SL2(R) representations for pretzel knots and shows left-orderability conditions.

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.

In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…

Study supports conjecture about pretzel links' homology.

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

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 b…

New knot groups found to be bi-orderable using pretzel knots.

It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…

Researchers study rational and pretzel knots using affine group representations.

New bounds found for ribbon numbers of knots and links.

We compute the reduced Khovanov homology of 3-stranded pretzel links. The coefficients are the integers with the "even" sign assignment. In particular, we show that the only homologically thin, non-quasi-alternating 3-stranded pretzels are P(-p,p,r) with p an odd integer and r greater than or equal to p (these were sho…

Study on folded ribbon knots and their minimum length.

We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with …