arXiv research

We prove that an iterated torus knot type fails the uniform thickness property (UTP) if and only if all of its iterations are positive cablings, which is precisely when an iterated torus knot type supports the standard contact structure. We also show that all iterated torus knots that fail the UTP support cabling knot …

Study calculates Kashaev invariants for twice-iterated torus knots.

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers …

Study calculates torsion for iterated torus knots using complex representations.

New family of knots with L-space surgeries but not algebraic.

In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pin points the error. In "An addendum on iterated torus knots" a more subtle result is proven resulting in giving a geometric realization of the Honda-Etny…

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…

We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.

New knots have unique Whitehead doubles.

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that $p/q$ is not a Jon…

Study shows $(p,q)$-cable knots cannot undergo certain types of surgery.

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If $M$ has a non-boundary-paralle…

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their constru…

In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved that all transversal knots whose topological knot type is that of an iterated torus knot (we call them cable knots) are transversally simple. That theorem is false, and the Erratum math.GT/0610565 identifies the gap. The purpose of this paper…

The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…

Paper confirms algebraic knots are linearly independent using twisted Blanchfield pairings.

Whitehead doubles have matching meridional rank and bridge number.

We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynom…

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellit…

Given a knot $K$ in $S^3$, a question raised by Cappell and Shaneson asks if the meridional rank of $K$ equals the bridge number of $K$. Using augmentations in knot contact homology we consider the persistence of equality between these two invariants under satellite operations on $K$ with a braid pattern. In particular…

Lower bounds on unknotting number for cabled knots.

New infinite families of twisted torus knots found.

New examples show algebraically slice knots with specific genus bounds.

Formula for Alexander polynomial of twisted torus knots derived.

Study shows differences of torus knots do not form $L$-space knots.

Study concordance of alternating torus knots to L-space knots.

New families of twisted torus knots found with essential surfaces.

New examples of positive twisted torus knots with same surface slope found.

The strong slope conjecture helps identify torus knots.

Study shows surgeries on certain knots bound rational homology 4-balls.

Knots formed from torus knots are not concordant to L-space knots.

Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.

Classifies Legendrian and transverse torus knots in $S^3$.

Study calculates fundamental groups of torus knots using algebraic topology.

Study finds infinite non-fibered twisted torus knots.

Study on sums of torus knots concordant to alternating knots.

Study shows surprising cobordism distances between certain torus knots.

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

Researchers confirm a relation between knot invariants and provide formulas for torus knots.

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relat…

Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.

Study shows unknotting number of certain virtual torus knots equals standard torus knot's unknotting number.

Characterizes Legendrian knots in lens spaces.

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…

Complexity class determined for recognizing torus knots.

Determine pairs of torus knots with genus one cobordisms, with exceptions.