arXiv research

The study explores knots and manifolds, proving properties and non-left-orderable groups.

A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that t…

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with re…

Formula found for braid index of $n$-bridge braids.

We give an alternative proof of a result of Kobayashi and Saeki that every genus one $1$-bridge position of a non-trivial $2$-bridge knot is a stabilization.

The $(1,1)$-length of a knot is a braid group invariant equaling its level number.

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of …

This paper proves the Boyer-Gordon-Watson conjecture for 1-bridge braids.

A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal f…

We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non- trivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors.

The study improves genus 1 bridge number bounds for satellite knots.

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. We generalize their construction and calculate th…

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Natura…

We show that if $K$ is a knot in $S^3$ and $Σ$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $Σ$ via an isotopy is $n$. We also construct a knot with two different bridge spheres with $2n$ and $2n-1$ bridges respecti…

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. Afte…

Let $K= K(w,b,t)$ be a 1-bridge braid in a solid torus $V$, and let $γ$ be a $(p,q)$ curve on the torus $T = \partial V$ of the exterior $M_K$ of $K$. It will be shown that Dehn filling on $T$ along $γ$ produces a solid torus if and only if $p$ and $q$ satisfy one of four conditions determined by the parameters $(w,b,t…

Constructs taut foliations for surgeries on positive 3-braids, confirming L-space conjecture.

We show that an $(n+1)$-bridge sphere for the unknot is a topologically minimal surface of index at most $n$.

A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However …

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namel…

The study connects twist positivity to L-space knots and concordance.

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…

Two proxy methods for causal identification are compared.

New parameterization for $(1,1)$-knots simplifies their study.

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surger…

The paper calculates the Hilbert polynomials for configuration spaces over graphs with a short circumference.

This paper concerns thin presentations of knots K in closed 3-manifolds M^3 which produce S^3 by Dehn surgery, for some slope gamma. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K …

The paper classifies a special family of knots in lens spaces using knot Floer homology.

Paper converts deep networks to flat, equivalent kernel machines.

Unified framework connects deformation theory and derived categories for multiparameter persistence.

Extends positive and almost positive links to successively almost positive ones.

The paper extends positivity results from vector bundles to Kobayashi positive ones.

The study establishes conditions for positive and quasi-positive links.

The paper defines new types of positivity and proves properties of Schur forms for vector bundles.

New characterizations of partial positivity using Hörmander's $L^2$-estimate.

Introduces $Θ$-positivity in Lie groups, generalizing Lusztig's positivity.

Uniform RC-positivity results for direct image bundles.

New bounds on Jones polynomial positivity for specific links.

We introduce the notion of a positive opetope and positive opetopic cardinals as certain finite combinatorial structures. The positive opetopic cardinals to positive-to-one polygraphs are like simple graphs to free omega-categories over omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit combinato…

Characterizes a subset of links using quasipositive and homogeneous properties.

Short note proves Brown-York mass positivity with new boundary conditions.

Positive braid knots have simple knot Floer homology.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

Extends Perelman's theorem to positive intermediate curvature conditions.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we p…

Satellite links of fully positive braids are characterized.

New $Θ$-positive representations of surface groups discovered.