Ichiro Torisu
We analyze transverse doubled knots in the standard contact 3-space by using spanned clasp disks. As applications, we will estimate their self-linking number and furthermore we will show that in many cases, transverse twist knots with the maximal self-linking number are unique up to transverse isotopy.
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A partial classification of 3-dimensional clasp number two, genus two fibered knots
本文在假设条件下的情况下，对亏格为二、clasp数为二的纤维化结进行了部分分类。
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather than out of crossings. We describe an equivalence relation on clasp diagrams which produces the isotopy classes of knots as equivalence classes. This equivalence relation is generated by local moves similar to the Reidemeister moves. Clasp diagrams produce particularly simple Seifert surfaces for knots and lead to an explicit formula for the Alexander-Conway p…
The clasp number $c(K)$ of a knot $K$ is the minimum number of clasp singularities among all clasp disks bounded by $K$. It is known that the genus $g(K)$ and the unknotting number $u(K)$ are lower bounds of the clasp number, that is, $\max\{g(K),u(K)\} \leq c(K)$. Then it is natural to ask whether there exists a knot $K$ such that $\max\{g(K),u(K)\}<c(K)$. In this paper, we prove that there exists an infinite family of prime knots such that the question above is affirmative.
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.
Chern-Simons functional, singular instantons, and the four-dimensional clasp number
本文利用源自等变奇异瞬子理论并与陈-西蒙斯泛函相关的纽结不变量，肯定回答了四维 clasp 数与切片亏格之差可否任意大的问题，同时解决 Livingston 猜想，并研究奇异瞬子 Frøyshov 不变量及其在无向切片亏格、环面纽结表示和协边中的应用。
Signed clasp numbers and four-genus bounds
本文研究了四维亏格与夹紧数的关系，通过构造新的例子展示了四维亏格可以无限大，扩展了之前的结果。
Slice genus, $T$-genus and $4$-dimensional clasp number
本文推广了切片结的$T$-亏格刻画，为切片亏格提供了$3$维刻画，证明$T$-亏格控制$4$维正交叉数，引入带状$T$-亏格并探讨其与切片结、带状结及着色链环的关联。
We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.
Recently Swatee Naik and Theodore Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called Strong S-equivalence, and prove that two oriented, ordered links L and L' are related by a sequence of doubled-delta moves if and only if they are Strongly S-equivalent. We also show that this is equivalent to the fact that L' can be obtained from L through a sequence of Y-clasper s…
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…
Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.
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…
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…
Doubled Disks and Satellite Surfaces
本文研究切片圆盘的正怀特黑德加倍与光滑同位素的关系，证明多种同调阻碍在卫星操作下保持，并据此构造出四维球中大量奇异圆盘及首个两两奇异切片圆盘的无限族。
Gordian distance and clasper surgery for links
本文证明，除了环绕数外，任何链接通过最多n^2次交叉变化可变为C_k平凡链接，从而表明Milnor不变量对解链数信息有限。
We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is applied to the theory of claspers for string links.
Diagram Systems and Generalized Finite Type Theories
本文以范畴论推广 Goussarov 定理，建立广义有限型理论生成集与拓扑对象图系统的对应，并用 clasp 图、looms 和虚拟横截结等例子说明其应用。
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