On the slice genus of knots
On the slice genus of knots
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关于结的切片属
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发表时间：
1982
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通讯作者：
P. Gilmer
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作者：
P. Gilmer
给定三维球面中的一个纽结K，K的亏格记为g（K），定义为K的Seifert曲面的最小亏格.切片亏格gs（K）定义为一个有向曲面G的最小亏格，它允许CG映射到K的4-球中的光滑真嵌入。如果我们坚持G的嵌入没有关于径向函数的局部极大值，我们得到了带状亏格gr（K）。因此，一个结是切片（带状）当且仅当gs（K）= 0（gr（K）= 0）。很明显，G。(K)<g，. (K)<=g（K）.存在由K的Seifert矩阵的不变量给出的gs（K）的已知下界。这些都包含在泰勒定义的不变量re（K）[8]中。它给出了基于塞弗特矩阵的最佳可能的界限，re（K）为零当且仅当塞弗特配对是代谢性的。如果是这种情况，K称为代数切片。卡森和戈登的工作[1，2，5]表明，某些代数切片结实际上不是切片。我们推广了[1]的主要定理。作为应用，我们给出了一个代数切片纽结序列Q，使得gs（Q，，）=g（Q，，）=n.我们还研究了图K，41=K的切片亏格，其中K，表示非纽结的t扭二重图。例如，我们证明g。（K12 =~ K1z）= 2。K I2是代数切片，但不是被[ l ]切片。第一节对链接形式作了一些说明。在第2节中，我们陈述并证明了我们的主要定理。在第三节中，我们给出了我们的例子。在本文中，所有的流形定向。我们用e表示欧拉特征线。
Given a knot K in the 3-sphere, the genus of K, denoted g(K), is defined to be the minimal genus for a Seifert surface for K. The slice genus gs(K) is defined to be the minimal genus of an oriented surface G admitting a smooth proper embedding in the 4-ball which maps CG to K. If we insist the embedding of G have no local maximum with respect to the radial function, we obtain the ribbon genus gr(K) instead. Thus a knot is slice (ribbon) if and only if gs(K) = 0 (gr(K)= 0). It is clear that g.~(K)<g,.(K)<=g(K). There are well known lower bounds on gs(K) given by invariants of a Seifert matrix for K. These are all included in the invariant re(K) [8] defined by Taylor. It gives the best possible bound based on a Seifert matrix, re(K) vanishes if and only if the Seifert pairing is metabolic. If this is the case, K is called algebraically slice. The work of Casson and Gordon [1, 2, 5] showed that certain algebraically slice knots are not in fact slice. We generalize the main theorem of [1]. As an application, we give a sequence of algebraically slice knots Q, such that gs(Q,)=g(Q,,)=n. We also study the slice genus of K, 41=K, where K, denotes the t twisted double of the unknot. We show for example that g.~(K 12 =~ K 1 z ) = 2 . K I2 is algebraically slice but not slice by [ l ] . Section 1 has some preliminaries on the linking form. In Sect. 2, we state and prove our main theorem. In Sect. 3 we give our examples. In this paper, all manifolds are oriented. We use e to denote the Euler characteristic.