Tetsuya Ito
We show that if a link $L$ has a closed $n$-braid representative admitting non-degenerate exchange move, an exchange move that does not obviously preserve the conjugacy class, $L$ has infinitely many non-conjugate closed $n$-braid representatives.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Positive permutation braids on n strings, which are defined to be positive n-braids where each pair of strings crosses at most once, form the elementary but non-trivial building blocks in many studies of conjugacy in the braid groups. We consider conjugacy among these elementary braids which close to knots, and show that those which close to the trivial knot or to the trefoil are all conjugate. All such n-braids with the maximum possible crossing number are also shown to be conjugate. We note that conjugacy of these braids for n<6 depends only on the crossing number. In contrast, we exhibit tw…
We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.
A generalization of the topological fundamental group is developed in order to exhibit a topologically complete braid group containing Artin's braid group on infinitely many strands with respect to the following notion of convergence: A sequence of braids b(n) converges to the trivial braid iff for each M>0 eventually the first M strands of b(n) are trivial.
We present a generalization of the Dehornoy-Brin braided Thompson group $BV_2$ that uses recursive braids. Our new groups are denoted by $BV_{n,r}(H)$, for all $n\geq 2,r\geq 1$ and $H \leq \mathcal{B}_n$, where $\mathcal{B}_n$ is the braid group on $n$ strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that $BV_{n,r}(H)$ is finitely generated if $H$ is finitely generated.
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.
An infinitary version of braid groups has been considered as a direct limit of n-braid groups. However, we can imagine more complicated braids with infinitely many strings. We invetisgate basic properties especially when the number of strings is countable.
Positive Braid Links with Infinitely Many Fillings
该研究证明了非同痕于标准有限型链环的正辫Legendrian链环存在无穷多精确拉格朗日填充，推进了相关拓扑领域研究。
Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fiber…
Non-braid positive hyperbolic $L$-space knots
本文研究非辫正超流体L-空间结，构造了无限多个此类结，证明其非辫正性，与Baker和Kegel的例子不同。
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.
The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface $F$ in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, $F$ is circular-foliated in a sense of Birman-Menasco's Braid foliation theory. As an application of the result, we prove that if Dehornoy floor of braids are larger than three, Nielsen-Thurston classification of braids and the geometry of their closure's complements are in one-to-one correspondence. Using this result, we construct infinitely many hyper…
A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.
We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our e…
We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz cla…
Knots whose braided satellite have the same HOMFLY polynomial up to given $z$-degrees
该研究针对给定纽结与正整数w，构造无穷多互不相同的双曲纽结，使它们与原纽结的对应辫状卫星HOMFLY多项式在指定z次内一致，推进了纽结多项式相关的拓扑学研究。
For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examples of surface links with free abelian group of rank two, concluding that their link types are infinitely many.
Satellite Links with Multiple Full Twists and Single-Twist Companions
研究卫星链环的正辫表示中全扭转数与其伴随链环的关系，构造了无限多卫星链环，其正辫表示具有任意多个全扭转，而伴随链环的正辫表示最多只有一个全扭转。
It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable…
A famous result of Bennequin states that for any braid representative of the unknot the Bennequin number is negative. We will extend this result to all n-trivial closed n-braids. This is a class of infinitely many knots closed under taking mirror images. Our proof relies on a non-standard parametrization of the Homfly polynomial.
Generating Infinitely Many Hyperbolic Knots with Plats
本文借助 plat 位置、辫群动力学与二重分支覆盖的 Heegaard 距离，利用 pseudo-Anosov 辫构造无穷多互不相同的素双曲 n-桥纽结，并给出亏格与体积的线性下界。
We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LG^{m,1}, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Delta. This result is not obvious since in the reduction, the representation of the braid group generator used to define LG^{m,1} does not satisfy a second-order characteristic identity unless m=1. To demonstrate that the one-variable reduction of LG^{m,1} satisfies the defining skein relation of Delta, we evaluate the kernel of a quantum trace…
The closure of a braid in a closed orientable surface $Σ$ is a link in $Σ\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.
Saddle point braids of braided fibrations and pseudo-fibrations
本文研究了首一复多项式环路中根与临界点构成的辫 $B_1$ 与 $B_2$ 间的关系，证明在特定条件下伪纤维化映射可成为纤维化映射，并由此构造出具有指定奇点链环的混合多项式。
Augmentations, Fillings, and Clusters
该研究用弗洛尔理论证明正辫子勒让德链环的增广簇是簇K2簇，其大量正合拉格朗日填充对应簇种子，还解决了非ADE型正辫子勒让德链环的无限填充问题。
For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard generators of B_n. This gives rise to a "comultiplication" map on link Floer homology. We use this to generate infinitely many new examples of prime topological link types which are not transversely simple.
Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms …
Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dim…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。