Jun O'Hara
We study a 1-form which can be given by a vector in a conformally invariant way. We then study conformally invariant functionals associated to a ``Y-diagram'' on the space of knots which are made from the 1-form.
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We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.
We construct a map from knots to (abstract) 2-knots which can be extended to higher dimensions; this map is the natural "knot" counterpart for "braid" theory of groups $G_{n}^{k}$.
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 the distortion of the (2,q)-torus knot is not bounded linearly from below.
We give an obstruction to unknotting a knot by adding a twisted band, derived from Heegaard Floer homology.
A note on longitudes of virtual knots
该研究指出经典纽结经度的换位子群性质可推广至虚拟纽结，为虚拟纽结的相关研究提供了新的结论
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.
Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.
We introduce a numerical invariant β(K) of a knot K which measures how non-alternating K is. We prove an inequality between β(K) and the (knot Floer) thickness of K. As an application we show that all Montesinos knots have thickness at most one.
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.
We show that every canonical Seifert surface is (up to isotopy) given by a knot diagram in which the (open) Seifert disks are pairwise disjoint.
We consider the question of when a slice knot admits a reducible Dehn surgery. By analyzing the correction terms associated to such a surgery, we show that slice knots cannot admit surgeries with more than two summands. We also give a necessary Heegaard Floer theoretic condition for a positive cable of a knot to be slice.
We prove that knot Floer homology of a certain class of knots is non-trivial in next-to-top Alexander grading. This gives a partial affirmative answer to a question posed by Baldwin and Vela-Vick which asks if the same is true for all non-trivial knots in $S^3$.
We compute the involutive knot invariants for pretzel knots of the form P(-2,m,n) for m and n odd and greater than or equal to 3.
It has been known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot K and any natural number n there exists a quasipositive knot Q whose Vassiliev invariants of order less than or equal to n coincide with those of K.
We show that a knot whose minimum crossing number $c(K)$ is even and greater than $30$ is not fertile; there exists a knot $K'$ with crossing number less than $c$ such that $K'$ is not obtained from a minimum crossing number diagram of $K$ by suitably changing the over-under information.
We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery as long as the JSJ piece of knot exterior does not contain $(2,r)$-torus exterior. We also show that an iterated torus knot other than $(2,p)$-torus knot does not admit chirally cosmetic surgery.
In this paper we clarify an issue in the knot surgery construction of Fintushel and Stern. Using knot surgery, they construct an infinite number of smooth structures on 4-manifolds satisfying certain conditions, but they do not explicitly work out the circumstances under which two manifolds that arise from their construction will fail to be diffeomorphic on the grounds of Seiberg-Witten theory. This paper fills in that gap.
Kawauchi proved that every strongly negative amphichiral knot $K \subset S^3$ bounds a smoothly embedded disk in some rational homology ball $V_K$, whose construction a priori depends on $K$. We show that $V_K$ is independent of $K$ up to diffeomorphism. Thus, a single 4-manifold, along with connected sums thereof, accounts for all known examples of knots that are rationally slice but not slice.
A note on the unknotting number and the region unknotting number of weaving knots
本文定义了辫图的翘曲度，通过对编织结图的翘曲度进行图解与组合检查，给出了某些编织结族的解结数和区域解结数的上界。
The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, and concordance genus.
A note on the involutive concordance invariants for certain (1,1)-knots
该研究完成了10交叉和11交叉(1,1)-纽结的对合协变不变量的计算工作，为相关纽结理论研究提供了具体的计算结果支撑。
For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.
Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by Hedden, Kim, and Livingston to prove $\mathbb{Z}_2^{\infty} < \mathcal{T}$ can be used to see $\mathbb{Z}_2^{\infty} < \mathcal{T}_0/\mathcal{T}_1$.
We give infinitely many examples of manifold-knot pairs (Y, J) such that Y bounds an integer homology ball, J does not bound a non-locally-flat PL-disk in any integer homology ball, but J does bound a smoothly embedded disk in a rational homology ball. The proof relies on formal properties of involutive Heegaard Floer homology.
We give an alternative proof of a recent theorem of Tange using the technology of changemaker lattices. Specifically, for $K\subset S^3$ a non-trivial knot with a lens space surgery, we give constraints on the Alexander polynomial of $K$ and the lens space surgery when the coefficient of $T^{g(K)-2}$ in $Δ_K(T)$ is non-zero.
Using an argument of Baldwin--Hu--Sivek, we prove that if $K$ is a hyperbolic fibered knot with fiber $F$ in a closed, oriented $3$--manifold $Y$, and $\widehat{HFK}(Y,K,[F], g(F)-1)$ has rank $1$, then the monodromy of $K$ is freely isotopic to a pseudo-Anosov map with no fixed points. In particular, this shows that the monodromy of a hyperbolic L-space knot is freely isotopic to a map with no fixed points.
A note on knot Floer homology of satellite knots with (1,1)-patterns
本文证明对于任意(1,1)-模式结P和伴随结K，卫星结P(K)的约化结Floer同调维数不小于P(U)和K的相应维数。
Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterize those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.
A note on alternating knots in handlebodies
本文研究手柄体中交替结的最小交叉数和旋转数，并推广Jones多项式到该设置中。
A note on the knot Floer homology of freely 2-periodic knots and their quotients
研究自由2-周期纽结P及其射影商纽结的纽结Floer同调，利用Lagrangian Floer同调中阶2作用的定位谱序列得到秩不等式，并推出Seifert genus的不等式。
Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in a paper in which he gives a complete classification of genus one fibered knots contained in lens spaces. Combining Baker's classification with Moser's characterization of lens space surgeries on the trefoil, we generate an infinite family of lens spaces which do not contain any …
A note on the X-torsion order of a knot
本文证明可用约化 Bar-Natan–Lee–Turner 谱序列计算由广义 Lee 复形定义的纽结 X-挠阶，并据此完成大量计算，给出阶为 4 的例子。
A note on cables and the involutive concordance invariants
本文建立了缆线结对合协边不变量的计算公式，并由此证明当结对合不变量非零时其（奇数，1）型迭代缆线非光滑切片，同时得到更强的解结数下界。
We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and the energies must be correctly regularized. We determine an explicit error bound between the energies in terms of the number of the edges of the polygon and the Ropelength of the smooth curve.
A note on knot fertility II
该研究给出纽结(m,n)-肥沃性的障碍条件，证明了(c(K)+f,c(K)+p)-肥沃纽结的有限性，并探讨最小交叉图的Seifert圈数与拧数，推进了纽结肥沃性相关理论研究。
We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than $11$ crossings except for three $10$-crossing knots and five $11$-crossing knots. We also compute the finite type invariant of order $3$ for two-bridge knots and Whitehead doubles, from which we prove several nonexistence results of purely cosmetic surgery.
We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.
Notes on constructions of knots with the same trace
本文用Gompf和Miyazaki的可对偶模式解释了Abe等人构造同迹纽结的操作$(*m)$，并证明Teragaito的同4-手术纽结族也可由此操作解释。
Errata: A note on orientation-reversing distance one surgeries on non-null-homologous knots
本文是对非零调纽结上反向距离一手术相关论文的勘误，指出并报告了原文献中的一个错误。
Lecture notes on link homologies and knotted surfaces
本文基于2024年Georgia拓扑暑期学校讲义，介绍链环同调理论如何通过配边同态研究四维空间中的打结曲面，并重点讲解Khovanov同调、Bar-Natan同调的计算技术及Bar-Natan范畴的作用。
Dai, Hom, Stoffregen and Truong defined a family of concordance invariants $φ_j$. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.
Hom gives an example of a knot with vanishing Upsilon invariant but nonzero epsilon invariant. We build more such knots that are linearly independent in the smooth concordance group.
These are the lecture notes for a course on Heegaard Floer homology held at PCMI in Summer 2019. We describe Heegaard diagrams, Heegaard Floer homology, knot Floer homology, and the relationship between the knot and 3-manifold invariants.
We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed $3$-manifolds. We first prove that, given $Y^3 \neq S^3$, for any non-trivial element $g\in π_1(Y)$ there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. In particular we consider these distinct smooth almost-concordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PL-disk in the Mazur manifold, but all the representatives we construct are topologically sli…
This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice genus at most 1. Further, we show that the lower bounds on the slice genus coming from the Tristram-Levine and Casson-Gordon signatures canno…
Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer $d$-invariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that $T(2, 5)$ is the only torus knot $T(2, m)$ with $m$ squa…
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.
A note on lattice knots
该研究针对格点纽结的Turing基本操作展开，发现其操作集合可被简化为仅一类简单的局部切换操作，为格点纽结相关操作的简化研究提供了新观察结论。
In the present note, we will show that there are infinitely many composite twisted torus knots.
Ozsvath-Stipsicz-Szabo recently defined a one-parameter family, upsilon of K at t, of concordance invariants associated to the knot Floer complex. We compare their invariant to the {-1, 0, 1}-valued concordance invariant epsilon, which is also associated to the knot Floer complex. In particular, we give an example of a knot K with upsilon uniformly equal to zero but epsilon non-zero.
Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S3, k) from a thin position of k. In this article, we will prove that if a thin position of k induces a thin decomposition of double branched cover of (S3,k) by Howards and Schultens' method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of k on top of a thin position of another prime summand, and so on. Therefore, k holds the nearly additivity of knot width (i.e. for k = k1#k2, w(k) = w(k1…
This paper has been withdrawn by the authors, as it was combined with "Conformally invariant energies of knots I" (math/0409396) to be "Conformally invariant energies of knots" which has replaced the former.
We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3--sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3--manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also intr…
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant $ν^+$: we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.
We compare the smooth concordance invariants Upsilon, phi and epsilon. Previous work gave examples of knots with one of the Upsilon and phi invariants zero but the epsilon invariant nonzero. We build an infinite family of linearly independent knots with both the Upsilon and phi invariants zero but the epsilon invariant nonzero.
Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some constructions related to foliations. We also interpret a theorem of Kauffman in the world of knot Floer homology, hence we can compute the top filtration term of the knot Floer homology for alternative links.
Knot quandles distinguish Suciu's ribbon knots
本文研究Suciu带状n-纽结的纽结quandle，证明尽管这些纽结的纽结群同构，其纽结quandle仍彼此不同构，并进一步计算了这些quandle的类型。
Circles Minimize most Knot Energies
该研究定义了一类新的重整化纽结能量，证明多数此类能量及O'Hara的多数纽结能量由正圆唯一最小化，证明了两个相关猜想，还找到非正圆最小化的能量，证明基于Luko的闭合曲线平均弦长定理。
In this note, we prove that a nontrivial Kinoshita-Terasaka or Conway knot does not admit chirally cosmetic surgeries, by calculating the finite type invariant of order 3.
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