增厚曲面中的充分链环与广义Tait猜想
Adequate links in thickened surfaces and the generalized Tait conjectures
AI总结：
本文应用Kauffman括号辫代数发展增厚曲面中的辫充分链环理论，证明了交错链环图是辫充分的，并由此建立第一和第二Tait猜想，推广了关于辫括号跨度与交叉点数的经典不等式及等号判定结果。
摘要：
本文应用Kauffman括号辫代数，发展了增厚曲面中辫充分链环的理论。我们证明了曲面上的任何交错链环图都是辫充分的。我们应用该理论为增厚曲面中的充分链环建立了第一和第二Tait猜想。我们的辫充分性概念比先前针对曲面上链环图所考虑的相应充分性概念更广泛且更强大。对于最小亏格为g(Σ)的曲面Σ上的链环图D，我们证明了$${\rm span}([D]_Σ) \leq 4c(D) + 4 |D|-4g(Σ),$$其中$[D]_Σ$是其辫括号，$|D|$是D的连通分支数，c(D)是交叉点数。这推广了Kauffman、Murasgi和Thistlethwaite的经典结果。我们进一步证明，当且仅当D是弱交错的，上述不等式取等号。这是Thistlethwaite关于经典链环著名结果的推广。因此，辫括号能检测弱交错链环的交叉点数。作为应用，我们证明了增厚曲面中充分链环在连通和下交叉点数是可加的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
