Unknotting the spun $T^{2}$-knot of a classical torus knot
Unknotting the spun $T^{2}$-knot of a classical torus knot
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解开经典环面结的旋转 $T^{2}$-结
DOI：
10.18910/23425
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发表时间：
2012
影响因子：
0.4
通讯作者：
Inasa Nakamura
中科院分区：
文献类型：
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作者：
Inasa Nakamura
证明了对于满足一定条件的经典辫的闭包，经典纽结的spunT 2-纽结的解纽数为1.本文给出了经典环面纽结的自旋T2-纽结的解结数为1的另一种证明。0.介绍一个表面knotis的图像的一个光滑的嵌入到欧氏4空间R4的一个封闭的连接表面。Kanenobu和Marumoto [10]证明了经典环面结的自旋2-结具有解结数1。因此，它遵循的spunT 2-结的经典环面结有解开号码一。这里，经典纽结K的自旋T2-纽结是K在3-球B3中与圆S1的乘积，通过B3 S1到R4的自然嵌入嵌入到R4中（[15，2]）。本文证明了对于满足一定条件的经典辫的闭包，经典纽结的spunT 2-纽结的解纽结数为1（定理3.1）.定理3.1给出了上述事实的另一种证明，即经典环面纽结的自旋T2-纽结具有解纽结数1（推论3.2）。定理3.1的证明是用图解法来表示的，即用一个表示自旋T2-纽结的表面链接图来表示。曲面链接图是一种在两个圆盘上的有限图，它有许多附加数据（[5，8，9]）。任何有向的曲面节点都由曲面链接图表示（[7，8，9]）。一个不打结的表面结由一个不打结的图表表示（[ 5，9]）。已知[4]，任何定向表面结S都可以通过沿有限数量的相互分离的1-手柄沿着应用1-手柄手术而变形为未打结的表面结。S的解结数是使S变形为解结所需的1-手柄的最小数量。无边是图表中的一条边，其端点是度为1的顶点。通过向呈现S的表面链接图添加自由边，沿沿着漂亮的单手柄将单手柄手术应用于定向表面结S（[6]）。2010年数学学科分类。小学57 Q45;中学57 Q35。
We show that for the closure of a classical braid which satisfi es certain conditions, the spunT2-knot of the classical knot has the unknotting number one. Th is gives an alternative proof of the fact that the spun T2-knot of a classical torus knot has the unknotting number one. 0. Introduction A surface knotis the image of a smooth embedding of a closed connected surface into the Euclidean 4-space R4. Kanenobu and Marumoto [10] showed that the spun 2-knot of a classical torus knot has the unknotting numb er one. Hence it follows that the spunT2-knot of a classical torus knot has the unknotting number one . Here, thespun T2-knot of a classical knotK is the product ofK in a 3-ball B3 with a circle S1, embedded intoR4 via the natural embedding of B3 S1 into R4 ([15, 2]). In this paper, we show that for the closure of a classical brai d which satisfies certain conditions, the spunT2-knot of the classical knot has the unknotting number one (Th eorem 3.1). Theorem 3.1 gives an alternative proof of the abov e-mentioned fact that the spun T2-knot of a classical torus knot has the unknotting number one (Corollary 3.2). The proof of Theorem 3.1 is shown by a diagrammatic method, by using a surface link chart presenting the spun T2-knot. A surface link chart is a sort of finite graph in a 2-disk with so me additional data ([5, 8, 9]). Any oriented surface knot is presented by a surfa ce link chart ([7, 8, 9]). An unknotted surface knot is presented by an unknotted chart ([ 5, 9]). It is known [4] that any oriented surface knot S can be deformed to an unknotted surface knot by applying 1-handle surgeries along a finite number of mutually disjoin t riented 1-handles. The unknotting numberof S is the minimum number of such 1-handles necessary to deform S to be unknotted. Afree edgeis an edge in a chart such that the end points are vertices of degree one. Applying a 1-handle surgery to an ori ented surface knot S along a nice 1-handle is presented by adding a free edge to a surface link chart presenting S ([6]). 2010 Mathematics Subject Classification. Primary 57Q45; Sec ondary 57Q35.
DOI：
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发表时间：
2013
期刊：
影响因子：
--
作者：
S. Imajo;et al;Seiichi Kamada
通讯作者：
Seiichi Kamada