论文标题
在增厚的表面和广义的泰特猜想中的足够链接
Adequate links in thickened surfaces and the generalized Tait conjectures
论文作者
论文摘要
在本文中，我们应用曲棍球手支架代数来开发一种在增厚表面中的绞线适当链接的理论。我们表明，表面上的任何交替链路图都足够了。我们应用理论来建立第一个和第二个泰特猜想，以在增厚表面中的足够联系。我们对绞线充足性的概念比以前考虑到表面上的链接图的相应的充分概念更广泛，更强大。 对于链接图$ d $在表面上的$σ$最小属$ g（σ）$的$，我们表明$$ {\ rm span}（\ rm span}（[d]_σ）\ leq 4c（d） + 4c（d） + 4 | d | d | -4g（σ） $ c（d）$是交叉数的数量。这扩展了考夫曼，穆拉斯吉和thistlethwaite的经典结果。我们进一步表明，当且仅当$ d $交替交替时，上述不平等是平等的。这是由于Thistlethwaite而导致的经典联系的众所周知结果的概括。因此，绞线支架检测到弱交替链路的交叉数。作为应用程序，我们表明，在连接的总和下，交叉数是加成的，以用于增厚表面中的足够链接。
In this paper, we apply Kauffman bracket skein algebras to develop a theory of skein adequate links in thickened surfaces. We show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate links in thickened surfaces. Our notion of skein adequacy is broader and more powerful than the corresponding notions of adequacy previously considered for link diagrams in surfaces. For a link diagram $D$ on a surface $Σ$ of minimal genus $g(Σ)$, we show that $${\rm span}([D]_Σ) \leq 4c(D) + 4 |D|-4g(Σ),$$ where $[D]_Σ$ is its skein bracket, $|D|$ is the number of connected components of $D$, and $c(D)$ is the number of crossings. This extends a classical result of Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if $D$ is weakly alternating. This is a generalization of a well-known result for classical links due to Thistlethwaite. Thus the skein bracket detects the crossing number for weakly alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces.
