arXiv research

The Conway knot can't be smoothly tied to any other knot an infinite number of times.

We construct an infinite order loose cork.

Here we give a concrete description of the cork automorphism $f:\partial W\to \partial W$ of the infinite order loose-cork $(W,f)$, defined in \cite{a2}. It is obtained by concatenating the defining ribbon disk of $W$ in $B^4$ by an infinite order isotopy of the boundary knot.

Infinite Klein bottles with 4-fold meridians found.

We prove the following well known conjecture: let $Σ$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $φ\in \textup{Mod}(Σ)$ be a an infinite order mapping class. Then there exists a finite solvable cover $\widehatΣ \to Σ$, and a lift $\widehatφ$ of $φ$ such that the action…

New invariant detects infinite order cabled knots.

We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-six of the sixty-seven knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order.

Suppose that $X,X'$ are simply-connected closed exotic 4-manifolds. It is well-known that $X'$ is obtained by an order 2 cork twist of $X$. We give an infinite exotic family of 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4…

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish tha…

We obtain the cohomology of the variational bicomplex on the infinite order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. This provides a solution of the global inverse problem of the calculus of variations of finite order on fiber bundles.

Study on mapping class groups of infinite type surfaces.

Study on knot $7_4$ surgeries reveals infinite residue characteristics and infinite order points.

Develops control and observer methods for complex systems.

We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group. This provides an obstruction to classical knots being of order 4. In particular, th…

Study shows infinite order in mapping class groups for certain 3D shapes.

Proves left-orderability of mapping class groups of infinite-type surfaces.

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4-manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct powers of $f$ yields pairwise nondiffeomorphic manifolds. The present paper exhibits …

We construct a compact, contractible 4-manifold $C$, an infinite-order self-diffeomorphism $f$ of its boundary, and a smooth embedding of $C$ into a closed, simply connected 4-manifold $X$, such that the manifolds obtained by cutting $C$ out of $X$ and regluing it by powers of $f$ are all pairwise nondiffeomorphic. The…

We develop a second-order model for limit order books in a single scaling regime.

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebr…

Smooth figure-eight knot cables have infinite order.

Algorithm confirms non-order-preserving braids, proving infinite family not order-preserving.

Paper establishes sufficient condition for comparing linear combinations of infinite-mean risks.

New curves share invariant up to any fixed order.

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group. As one application, recall that the n-twisted double of an arbitrary knot has o…

Let $Σ$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $Σ$. We show that a Torelli mapping class either acts with infinite order on the homolo…

New 4-manifolds with exotic diffeomorphisms found.

Finite order elements with infinite centralizers in 3-manifold groups imply specific structure.

Every exotic pair in 4-dimension is obtained each other by twisting a {\it cork} or {\it plug} which are codimension 0 submanifolds embedded in the 4-manifolds. The twist was an involution on the boundary of the submanifold. We define cork (or plug) with order $p\in {\Bbb N}\cup \{\infty\}$ and show there exists a plug…

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conj…

Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.

FDPs generalize diffusion models to function spaces, enabling efficient image generation.

We construct an infinite order cork (W,f), which means that W is a smooth compact contractible 4-manifold with Stein structure, and f is a self diffeomorphism of the boundary of W, such that the n-fold composition maps f^{n}=f o f o... o f give rise to smoothly distinct corks (W, f^{n}) for sufficiently large values of…

The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite sub…

The paper finds infinite families of non-left-orderable L-spaces from tangles.

The paper finds infinitely many metrics with constant sixth order Q-curvature on spheres and related manifolds.

New knots show linear independence in slice concordance.

In the previous paper the author defined an infinite order plug $(P,\varphi)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are…

Proves existence of infinite order elements in curvature spaces.

Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…

Paper investigates optimal transport map estimation in infinite-dimensional spaces.

Study on disk configurations in strips shows stability patterns.

Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The challenge is compounded for infinite-order Markov processes, since conditioning on fi…

We show the homotopy spheres $Σ_{n} = -W\smile_{f^{n}}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f: \partial W \to \partial W$ is $S^4$. To do this we first show that $Σ_{n}$ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-han…

In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is genera…

This paper computes the quadratic Witt groups (the Wall L-groups) of the polynomial ring Z[t] and the integral group ring of the infinite dihedral group, with various involutions. We show that some of these groups are infinite direct sums of cyclic groups of order 2 and 4. The techniques used are quadratic linking form…

For any positive integer $n$ we give a ${\mathbb Z}^n$-cork with a ${\mathbb Z}^n$-effective embedding in a 4-manifold being homeomorphic to $E(n)$. This means that a cork gives a subset ${\mathbb Z}^n$ in the differential structures on $E(n)$. Further, we describe handle decompositions of the twisted doubles (homotopy…

Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.