存在长纽结的非同伦框架同伦
There are non homotopic framed homotopies of long knots
摘要：
设$\mathcal {M}$是3-空间中所有的空间，包括奇异的长纽结，并且对于该空间，到平面的固定投影是浸入。设$cl(Σ^{(1)}_{iness})$是$\mathcal {M}$中所有奇异结点的并集的闭包，其中恰好有一个普通的二重点，并且使得两个分解表示相同的（非奇异的）结点类型。我们称$Σ^{(1)}_{iness}$为{ \em非本质墙}，称$\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$为{ \em本质图空间}。通过对Kauffman括号的扩展，构造了$H^1(\mathcal {M}_{ess}; \mathbb{Z}[A, A^{-1}])$中的一个非平凡类.这特别意味着在$\mathcal {M}_{ess}$中有环，它由结的规则同位素和交叉变化组成，并且在$\mathcal {M}_{ess}$中是不可收缩的（导致论文的标题）。我们猜想，我们的建设产生了一个新的结多项式的unknotting号码1。
英文摘要：
Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(Σ^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call $Σ^{(1)}_{iness}$ the {\em inessential walls} and we call $\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$ the {\em essential diagram space}. We construct a non trivial class in $H^1(\mathcal {M}_{ess}; \mathbb{Z}[A, A^{-1}])$ by an extension of the Kauffman bracket. This implies in particular that there are loops in $\mathcal {M}_{ess}$ which consist of regular isotopies of knots together with crossing changings and which are not contractible in $\mathcal {M}_{ess}$ (leading to the title of the paper). We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one.
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