arXiv subjects

Tetsuya Abe

Publications and source records attributed to Tetsuya Abe.

14 recordsLinked to original sources

arXiv2024

Complexity of equal 0-surgeries

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincar\'e conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. In particular, we compute the minimum of c(K)+c(K') and of t(K)+t(K') among all friends K and K'. Along the way, we classify all 0-surgeries of prime knots of at most 15 crossings. Moreover, we determine for many friends in our census if their traces are equivalent or not. For that, we develop a new obstruction for two traces being homeomorphic coming from symmetry-exceptional slopes of hyperbolic knots. This is enough to also determine the minimum value of c(K)+c(K') among all friends K and K' whose traces are not homeomorphic.

math.GT↗

arXiv2023

Self-supervised Extraction of Human Motion Structures via Frame-wise Discrete Features

The present paper proposes an encoder-decoder model for extracting the structures of human motions represented by frame-wise discrete features in a self-supervised manner. In the proposed method, features are extracted as codes in a motion codebook without the use of human knowledge, and the relationship between these codes can be visualized on a graph. Since the codes are expected to be temporally sparse compared to the captured frame rate and can be shared by multiple sequences, the proposed network model also addresses the need for training constraints. Specifically, the model consists of self-attention layers and a vector clustering block. The attention layers contribute to finding sparse keyframes and discrete features as motion codes, which are then extracted by vector clustering. The constraints are realized as training losses so that the same motion codes can be as contiguous as possible and can be shared by multiple sequences. In addition, we propose the use of causal self-attention as a method by which to calculate attention for long sequences consisting of numerous frames. In our experiments, the sparse structures of motion codes were used to compile a graph that facilitates visualization of the relationship between the codes and the differences between sequences. We then evaluated the effectiveness of the extracted motion codes by applying them to multiple recognition tasks and found that performance levels comparable to task-optimized methods could be achieved by linear probing.

cs.CV↗

arXiv2022

Ribbon concordance and the minimality of tight fibered knots

Agol proved that ribbon concordance forms a partial ordering on the set of knots in the $3$-sphere. In this paper, we prove that all tight fibered knots are minimal in this partially ordered set. We also give the table of prime minimal knots up to $8$-crossings except for $8_{15}$.

math.GT↗

arXiv2020

Knots with infinitely many non-characterizing slopes

Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$, $8_3$, $8_4$, $8_6$, $8_7$, $8_9$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$,$8_{20}$ and $8_{21}$ have infinitely many non-characterizing slopes. We also introduce the notion of trivial annulus twists and give some possible applications. Finally, we completely determine which knots have special annulus presentations up to 8-crossings.

math.GT↗

arXiv2018

Ribbon disks with the same exterior

We construct an infinite family of slice disks with the same exterior, which gives an affirmative answer to an old question asked by Hitt and Sumners in 1981. Furthermore, we prove that these slice disks are ribbon disks.

math.GT↗

arXiv2017

Characterization of positive links and the $s$-invariant for links

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongly quasipositive. On the other hand, it implies that almost positive links are never homogeneous links.

math.GT↗

arXiv2016

Fibered knots with the same $0$-surgery and the slice-ribbon conjecture

Akbulut and Kirby conjectured that two knots with the same $0$-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.

math.GT↗

arXiv2015

A construction of slice knots via annulus twists

We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.

math.GT↗

arXiv2015

Infinitely many knots admitting the same integer surgery and a 4-dimensional extension

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, the first by a method of twisting along an annulus and the second by a generalization of this procedure. The latter family also solves a stronger version of Problem 3.6(D), that for any integer $n$, there exist infinitely many mutually distinct knots such that 2-handle addition along each with framing $n$ yields the same 4-manifold.

math.GT↗

arXiv2014

Annulus twist and diffeomorphic 4-manifolds II

We solve a strong version of Problem 3.6 (D) in Kirby's list, that is, we show that for any integer $n$, there exist infinitely many mutually distinct knots such that $2$-handle additions along them with framing $n$ yield the same $4$-manifold.

math.GT↗

arXiv2013

Annulus twist and diffeomorphic 4-manifolds

We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the $n$-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.

math.GT↗

arXiv2011

Unoriented band surgery on knots and links

We consider a relation between two kinds of unknotting numbers defined by using a band surgery on unoriented knots; the band-unknotting number and H(2)-unknotting number, which we may characterize in terms of the first Betti number of surfaces in S^3 spanning the knot and the trivial knot. We also give several examples for these numbers.

math.GT↗

arXiv2010

The Rasmussen invariant of a homogeneous knot

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader's theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.

math.GT↗

arXiv2008

The dealternating number and the alternation number of a closed 3-braid

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

math.GT↗