Marco Mackaay, Paul Turner, Pedro Vaz
We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
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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.
We present a large family of knots for which the Rasmussen s-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer-Mrowka and Cochran-Harvey-Horn.
We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a $(p,q,q)$-pretzel knot with integers $p,q \ge 2$ admits no Seifert fibered surgery.
The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov definition of Khovanov homology for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We then discuss the implications of the maps induced in the aforementioned theory to the…
The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal
本文证明了几乎正结的Rasmussen不变量、4亏格和3亏格相等，并基于其几乎正结图确定了Rasmussen不变量，得出任何几乎正结均非齐次且不存在4亏格为1的几乎正结。
We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.
We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus knot T(m,n) by a fixed constant for all but finitely many n>0. Combining this result together with Hedden's extensive work on the behavior of tau on (m,mr+1)-cables yields bounds on the value of tau on any (m,n)-cable of K. In addition, several of Hedden's obstructions for cables b…
Let νbe any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |ν(K)| <= g_4(K), and determines the 4-ball genus of positive torus knots, ν(T_{p,q}) = (p-1)(q-1)/2. Either of the knot concordance invariants of Ozsvath-Szabo or Rasmussen, suitably normalized, have these properties. Let D_{\pm}(K,t) denote the positive or negative t-twisted double of K. We prove that if ν(D_{+}(K,t)) = \pm 1, then ν(D_{-}(K,t)) = 0. It is also shown that ν(D_{+}(K,t))= 1 for all t <= TB(K) and ν(D_{+}(K, t)) = 0 for all t \ge -TB(-K), where TB(K) denotes the Thurston-Bennequin num…
We use simple properties of the Rasmussen invariant of knots to study its asymptotic behaviour on the orbits of a smooth volume preserving vector field on a compact domain in the 3-space. A comparison with the asymptotic signature allows us to prove that asymptotic knots are non-alternating, in general. Further we show that the Rasmussen invariant defines a quasi-morphism on the braid groups and derive estimates for the stable commutator and torsion lengths of alternating braids.
Given a diagram D of a knot K, we give easily computable bounds for Rasmussen's concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.
We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Focusing on the case of 3-braids, we compute the Rasmussen $s$ invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $ψ$ invariant is entirely determined by the $s$ invariant and the self-linking…
We define a Rasmussen $s$-invariant over the coefficient ring of the integers, and show how it is related to the $s$-invariants defined over a field. A lower bound for the slice genus of a knot arising from it is obtained, and we give examples of knots for which this lower bound is better than all lower bounds coming from the $s$-invariants over fields. We also compare it to the Lipshitz-Sarkar refinement related to the first Steenrod square.
Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.
A diagrammatic approach to the Rasmussen invariant via tangles and cobordisms
提出基于Bar-Natan Khovanov同调的图解方法，从缠绕分解局部计算Rasmussen s-不变量，并应用于所有3股辫结。
As proved by Hedden and Ording, there exist knots for which the Ozsvath-Szabo and Rasmussen smooth concordance invariants, tau and s, differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice. It is shown in this note that a simple manipulation of the Hedden-Ording examples yields a topologically slice Alexander polynomial one knot for which tau and s differ. Manolescu and Owens have previously found a concordance invariant that is independent of both tau and s on knots of polynomial one, and as a consequence have shown that the smooth concordance…
A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball
本文为Khovanov同调的生成树复形明确定义了微分形式，描述了Rasmussen的s-不变量，并据此发现了一个在4-球中界定奇异圆盘的无限纽结族。
Stable deformed $\mathfrak{gl}_N$ homology of torus knots
利用链环分裂形变（$y$-ification）工具，计算并证实了环面纽结 $\mathfrak{gl}_N$ 稳定同调中 Rasmussen 谱序列的相关猜想。
We produce chain-level generators of the virtual Lee complex $ Kh ' (V ) $ and use them to convert the computable bounds on the Rasmussen invariant of classical knots due to Kawamura and Lobb into bounds on the virtual Rasmussen invariant as defined by Dye, Kaestner, and Kauffman. We also exhibit a class of diagrams for which the bounds are tight. In addition, we use the chain-level generators to show that the virtual Rasmussen invariant is additive with respect to connect sum.
Rasmussen invariants of Whitehead doubles and other satellites
本文借助Khovanov与Bar-Natan同调的多曲线技术，建立环绕数为2的卫星结的$\mathbb{F}_2$-Rasmussen不变量公式，并考察其与$τ$不变量的可能联系。
A virtual knot is an equivalence class of embeddings of $ S^1 $ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of $ S^1 $ into a thickened surface $ Σ_g \times I $; what is the minimal genus of oriented surfaces $ S \hookrightarrow M \times I $ with the embedded $ S^1 $ as boundary, where $ M $ is an oriented $ …
Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.
In this paper we present several counterexamples to Rasmussen's conjecture that the concordance invariant coming from Khovanov homology is equal to twice the invariant coming from Ozsv{á}th-Szab{ó} Floer homology. The counterexamples are twisted Whitehead doubles of the (2,2n+1) torus knots.
A Rasmussen invariant for links in $\mathbb{RP}^3$
本文通过构造 $\mathbb{RP}^3$ 中链环 Khovanov 同调的 Lee 型形变，定义了 Rasmussen s-不变量的类似物，并以此约束链环配边亏格，进而给出了 $S^3$ 中合痕但非标准等变合痕的自由 2-周期纽结实例。
We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.
Squeezed Knots
本文研究了压缩结的性质，通过量子不变量分析其压缩性，并探讨了相关同调操作的障碍。
We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.
A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpart of s(K) is not additive for connected sums.
Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.
In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stabili…
A proof of Dunfield-Gukov-Rasmussen Conjecture
本文通过构造从 gl_0 同调到纽结 Floer 同调的 Bockstein 型双分次谱序列，证明了 Dunfield-Gukov-Rasmussen 猜想，并推出相关同调可识别若干经典纽结。
We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.
Local equivalence and refinements of Rasmussen's s-invariant
本文受Floer同调局部等价概念启发，构造整合奇偶Khovanov同调的LEO三元组局部等价群，推导多个细化s不变量，通过实验验证其阻碍纽结切片性的能力。
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.
We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case and define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular it is an obs…
A new concordance homomorphism from Khovanov homology
本文利用纽结的泛Khovanov链复形构造了一个新的光滑配边群同态，并证明其严格强于所有不同特征域上Rasmussen不变量的组合。
The Conway knot has infinite concordance order
利用 Rasmussen 不变量、卫星操作和零调扭转，证明 Conway 结在光滑协和群中具有无限阶。
We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia and Haiman.
Diagonal knots and the tau invariant
本文证明所有可由对角网格图表示的纽结均为正纽结，并给出一个非环面纽结的实例，扩展了该表示类的已知范围。
Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are related to spanning trees of the Tait graph, and we exploit this result to investigate the structure of Khovanov homology for alternating knots. Further, we extend Rasmussen's invariant to links. Finally, we generalize Khovanov's categorifications of the colored Jones polynomial…
We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen's invariant s(K). We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of k…
The pretzel knot $P(4, -3, 5)$ is not squeezed
本文通过比较Rasmussen不变量与Iida、Taniguchi提出的$q_M$不变量，证明无穷族三股pretzel扭结非挤压扭结，特别证得$P(4, -3, 5)$非挤压扭结，解答了Lewark2024年提出的问题。
To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant $Υ^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's $h_i$ invariants, and the Ozsv\' ath-Stipsicz-Szab\' o upsilon invariant. Furthermore, to three such regions $C$, $C^+$ and $C^- $ we associate invariants $Υ_{C^\pm, C}$ generalising Kim-Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use th…
We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.
Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya's invariant in the odd Khovanov homology of Ozsváth, Rasmussen, and Szabó. We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya's invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and sel…
We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot $K$. As an application to connected sums, we prove that if a knot in the three-sphere admits an $L$-space surgery, it must be a prime knot. As an application of the computation of $d$-invariants, we show that the Alexander polynomial is a concordance invariant within the class of $L$-space knots, and show the four-genus bound given by the $d$-invariant of +1-su…
Involutive Khovanov homology and equivariant knots II
本文将对合Khovanov同调框架推广到对合缠结，实现了相关同调与不变量的分治加速计算，并得到存在奇异切片圆盘对的强可逆纽结新例子。
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsvath-Szabo-Rasmussen's tau-invariant to obstruct almost-concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost-concordant knots. Finally we …
Khovanov homology and refined bounds for Gordian distances
本文从Khovanov同调提取纽结Gordian距离的新下界，结合并强化了Rasmussen与挠不变量的结果，并改进了真有理Gordian距离的下界。
Involutive Khovanov homology and equivariant knots
本文提出了一种involutive Khovanov同调，用于研究强不可约结的exotic slice disks性质，并重新证明了Hayden的无限族结J_n的exotic性。
Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.
Let $ν$ be either the Ozsváth-Szabó $τ$-invariant or the Rasmussen $s$-invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_ν(K)$ to be the minimum of $k$ for which $ν$ of the $k$-twisted positive Whitehead double of $K$ vanishes. They proved that $t_ν(K)$ is bounded above by $-TB(-K)$, where $TB$ is the maximal Thurston-Bennequin number. We use a blowing up process to find a crossing change formula and a new upper bound for $t_ν$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between L…
We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These "relative adjunction inequalities" improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every n…
We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial.
In this paper we examine the relationship between various types of positivity for knots and the concodance invariant tau discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, tau characterizes strong quasipositivity. This is quantified by the statement that for K fibered, tau(K)=g(K) if and only if K is strongly quasipositive. In addition, we survey existing results regarding tau and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive. For instance, we show that any k…
Khovanov homology and exotic $4$-manifolds
通过Khovanov-Rozansky gl_2 skein lasagna模区分怪异结迹，首次证明了怪异紧致定向4维流形的存在，并提出了无法通过规范/弗洛尔理论方法恢复的怪异结迹家族。
We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb's conjecture that the sl_3-knot concordance invariant s_3 (suitably normalised) should be equal to the Rasmussen invariant s_2. For this family, |s_3| < |s_2|. However, we also find other knots for which |s_3| > |s_2|. The main tool is an implementation of Morrison and Nieh's algorithm to calculate Khovanov's sl_3-foam link homology. Our C++-program is fast enough to calculate the integral homology of e.g. the (6,5)-torus knot in six minutes. Furthermore, we propose a potent…
Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line
本文引入由三角划分索引的广义Schröder多项式，证明其与Coxeter结的Khovanov-Rozansky同调Poincaré级数一致，并借此证明了ORS猜想的特化及$(q,t)$-Schröder定理的三角版本。
Let $D_+(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed integer-valued additive concordance invariant $ν$ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $t_ν(K)$ to be the greatest integer $t$ such that $ν(D_+(K,t)) = 1$. Let $K_1$ and $K_2$ be any knots then we prove the following inequality : $t_ν(K_1) + t_ν(K_2) \leq t_ν(K_1 \# K_2) \leq min(t_ν(K_1) - t_ν(-K_2), t_ν(K_2) - t_ν(-K_1)).$ As an application we show that $t_τ(K) \neq t_s(K)$ for infinitely many knots and that their differenc…
A $y$-ification of Khovanov homology
本文通过构造Khovanov同调的$y$-化及其等变版本，定义了$\mathfrak{sl}_2$作用，并证明其与HOMFLY-PT同调到Khovanov同调的谱序列相容，从而区分具有相同同调的结对。
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