Jacob Mostovoy
A rope is a non-singular embedding of a closed interval into R^3, which sends the ends of the interval to some fixed points A and B such that |AB|=1. A rope is short if its length is less than 3. The main result of the paper is that the fundamental group of the space of short ropes is naturally isomorphic to the group completion of the semigroup of knots.
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We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a gr…
Ropelength-minimizing concentric helices and non-alternating torus knots
本文研究非交替环面纽结的同心螺旋结构，通过优化几何与组合特性，为3到39个螺旋系统找到高效构型，并为更大系统提出优化构造，将非交替环面纽结的ropelength上下界比率从29降至1.4-3.8。
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave bett…
We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twisting region, then it has a non-trivial knot and its mirror image as connected summands.
In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.
This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory.
In this short article I introduce the knotR package, which creates two dimensional knot diagrams optimized for visual appearance using the R programming language. The knotR package is a systematic R-centric suite of software for the creation of production-quality artwork of knot diagrams, released under GPL2.
In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.
We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which …
In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.
In this note, we give a short proof that knot Floer thickness is a lower bound on the dealternating number of a knot. The result is originally due to work of Abe and Kishimoto, Lowrance, and Turaev. Our proof is a modification of the Stipsicz-Szabo approach using Kauffman states to show that thickness bounds the minimal number of bad domains in a knot diagram.
Transverse fibered knots and Taut Foliations
本研究给出Honda-Kazec-Matic定理的简短替代证明，验证了满足特定条件的纤维纽结所支撑的切触结构可由紧叶状结构微扰得到，为相关几何拓扑结论提供了新的证明路径。
This paper is a review of the book "Knots" by Alexei Sossinsky. The review includes a short personal history of knot theory at the end of the twentieth century.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yields an L-space knot. Consequently, any knot obtained by a non-trivial band sum cannot admit a finite surgery. For context, we exhibit two examples of non-trivial band sums of tight fibered knots producing prime knots: one is fibered but not tight, and the other is strongly quasip…
This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the algebraic topology of the knot exterior, including Alexander invariants, the Farber-Levine pairing and Casson-Gordon invariants, as well as gauge theoretic and combinatorial invariants. Details are scarce and new results inexistant.
Transverse knots determined by their cyclic branched covers
本文研究横截结的横截同胚类，通过构造其他非同胚但循环分支覆盖接触同胚的横截结例子，证明许多横截结的横截同胚类由其循环分支覆盖的接触同胚类型决定。
This is survey about the classical knot concordance group, prepared for an upcoming handbook of knot theory. Topics include: the basic definitions of concordance; the theory of algebraic concordance as developed by Levine; the theory of Casson-Gordon invariants; applications of topological surgery as developed by Freedman; smooth results based on differential geometric techniques such as those of Donaldson; the filtration of the concordance group as developed by Cochran, Orr, and Teichner; and relations to classical 3-dimensional knot theory. There is also a short problem list.
Shortcut learning in geometric knot classification
本文研究了机器学习在几何结分类中的捷径方法，揭示训练数据中的非拓扑特征，并提供公开数据集和代码以促进ML在该领域的应用。
In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polynomial invariants of singular knots obtained from the singquandle structure. We then explore psyquandles which can be thought of as generalizations of oriented signquandles, and review recent developments regarding invariants of singular knots obtained…
Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space for all pairs of integers (r,s) except (0,0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz--Wu and Kang all fit this construction. As such, we propose a generalization of the Cabling Conjecture of Gonzales-Acuna…
Legendrian simple knots not detected by Khovanov homology
本研究构造了Khovanov同调无法检测的Legendrian简单纽结，推翻了Chernov和Maguire的相关猜想。
In this short note, we prove the cyclotomic expansion formula for the colored HOMFLY-PT invariant of double twist knots, it confirms the cyclotomic expansion conjecture for $SU(N)$-invariants proposed in \cite{CLZ}.
Quasipositivity and braid index of pretzel knots
本文利用Morton-Franks-Williams不等式和Khovanov-Rozansky concordance homomorphisms，计算了一条链索上具有偶数个交叉点的三股pretzel knots的braid index，并确定了其quasipositivity。
Triangulating Spun 2-Knot Complements
研究如何对通过旋转1 - 纽结得到的2 - 纽结外部进行三角剖分，核心方法是提出算法并在Regina中实现，主要贡献是给出了由至多八个交叉的1 - 纽结得到的外部的三角剖分。
In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a concordance from the trivial Legendrian knot with maximal Thurston-Bennequin invariant to itself. This allows to obstruct concordances from the Pretzel knot P(3,-3,-m) when m > 3 to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.
The first part of this paper is a short review of the construction [dg-ga/9710001] of invariants of rational homology 3-spheres and knots in terms of configuration space integrals. The second part describes the relationship between the above construction and Kontsevich's proposal of removing one point from the rational homology sphere. Explicit formulae are computed. In the case of the "Theta" invariant, a comparison with Taubes's construction is briefly discussed.
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