双变量Conway多项式与Cochran导出不变量
Two-variable Conway polynomial and Cochran's derived invariants
AI总结：
本文研究链环的双变量Conway多项式与Cochran导出不变量的关系，在环绕数为0和1两种情形下给出系数$α_{1,2k-1}$的公式，并用Seifert矩阵提供新证明。
摘要：
我们注意到，对于一个m分支链环L（m>1），其Conway势函数$Ω_L$可以表示为$Ω_L(x_1,\dots,x_m)=Θ_L(\nabla_L(x_1-x_1^{-1},\dots,x_m-x_m^{-1}))$，其中$\nabla_L\in\mathbb Z[z_1,\dots,z_m]$是唯一确定的，而$Θ_L$是加法群$\mathbb Z[x_1^{\pm1},\dots,x_m^{\pm1}]$的某个自同态，它仅依赖于L各分支之间的两两环绕数。受拓扑同痕应用的驱动，我们研究形式幂级数$\bar\nabla_L$，它由$\nabla_L$除以L各分支的Conway多项式得到。对于一个满足$lk(L)=0$的2分支链环，$\bar\nabla_L(u,v)$在$uv^{2k-1}$处的系数$α_{1,2k-1}$等于Cochran的导出不变量$(-1)^{k+1}β^k(L)$。虽然这一事实可以从G.-T. Jin的一个结果推出——他利用Alexander多项式的手术观点证明了该结果——但我们使用Seifert矩阵给出了另一种证明。我们的主要结果是在几何上更为精细的情形$lk(L)=1$下，同一系数$α_{1,2k-1}$的公式。具体而言，我们用广义Cochran不变量$β_F^{ij}(P,Q)$来表示它，这些不变量由Gilmer--Livingston（当$P=Q$时）和Tsukamoto--Yasuhara（当$j=0$时）研究过，并且与纽结无限循环覆盖中的Cochran配对密切相关。
英文摘要：
We note that the Conway potential function $Ω_L$ of an $m$-component link $L$, $m>1$, can be expressed as $Ω_L(x_1,\dots,x_m)=Θ_L(\nabla_L(x_1-x_1^{-1},\dots,x_m-x_m^{-1}))$ for a unique $\nabla_L\in\mathbb Z[z_1,\dots,z_m]$, where $Θ_L$ is a certain endomorphism of the additive group of $\mathbb Z[x_1^{\pm1},\dots,x_m^{\pm1}]$ which depends only on the pairwise linking numbers of the components of $L$. Motivated by applications to topological isotopy, we study the formal power series $\bar\nabla_L$, obtained by dividing $\nabla_L$ by the Conway polynomials of the components of $L$. For a $2$-component link with $lk(L)=0$, the coefficient $α_{1,2k-1}$ of $\bar\nabla_L(u,v)$ at $uv^{2k-1}$ equals Cochran's derived invariant $(-1)^{k+1}β^k(L)$. While this can be deduced from a result of G.-T. Jin, which he proved using the surgical view of the Alexander polynomial, we provide an alternative proof, using Seifert matrices. Our main result is a formula for the same coefficient $α_{1,2k-1}$ in the geometrically subtler case $lk(L)=1$. Namely we express it in terms of generalized Cochran invariants $β_F^{ij}(P,Q)$, which were studied by Gilmer--Livingston (when $P=Q$) and by Tsukamoto--Yasuhara (when $j=0$) and are closely related to the Cochran pairing in the infinite cyclic covering of a knot.
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