Homological thickness and stability of torus knots
Homological thickness and stability of torus knots
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环面结的同源厚度和稳定性
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发表时间：
2005
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通讯作者：
Marko Stosic
中科院分区：
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作者：
Marko Stosic
在本文中，我们证明了非交变环面结是同构厚的，即它们的Khovanov同构至少占据三条对角线。进一步证明了在不改变环面结的某一部分同源性的情况下，可以减少环面结的全扭转数，从而证明了由Dunfield、Gukov和Rasmussen在cite{dgr}中推测的环面结存在稳定的同源性。由于我们的主要工具是同调中的长精确序列，我们将我们的方法应用于Khovanov-Rozansky （$sl(n)$）同调的情况，从而得到了环面结的$sl(n)$同调的类似稳定性性质，也在引用{dgr}中推测。
In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in cite{dgr}.