随机空间与平面曲线
Random space and plane curves
摘要：
我们研究随机结，我们将其定义为随机周期函数的三元组（其中随机函数是随机三角级数，具有$a_k, b_k$的\[f(θ) = \sum_{k=1}^\infty a_k \cos (k θ) +b_k (\sin k θ),\]是具有平均$0$和方差$σ(k)^2$的独立高斯随机变量-我们的结果将取决于$σ$对$k.$的函数依赖性。我们表明，如果$σ(k) = k^α,$与$α< -3/2,$，则得到允许具有$N$交叉的投影的结类型的概率至少与$1/N.$一样快地衰减。常数$3/2$是显著的，因为有$α< -3/2$正好是$f(θ)$成为$C^1$函数的条件，所以我们的类正好是随机\emph{tame}结的类。我们还发现了一些令人惊讶的实验观察亚历山大多项式的随机结（缓慢和非衰减系数）的零点，甚至更令人惊讶的观察其系数。我们的观察结果在其他随机节点模型中也存在，这使得结果可能具有普遍性。
英文摘要：
We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(θ) = \sum_{k=1}^\infty a_k \cos (k θ) +b_k (\sin k θ),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $σ(k)^2$ - our results will depend on the functional dependence of $σ$ on $k.$ In particular, we show that if $σ(k) = k^α,$ with $α< -3/2,$ then the probability of getting a knot type which admits a projection with $N$ crossings, decays at least as fast as $1/N.$ The constant $3/2$ is significant, because having $α< -3/2$ is exactly the condition for $f(θ)$ to be a $C^1$ function, so our class is precisely the class of random \emph{tame} knots. We also find some suprising experimental observations on the zeros of Alexander polynomials of random knots (with slowly and non-decaying coefficients), and even more surprising observations on their coefficients. Our observations persist in other models of random knots, making it likely that the results are universal.
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。
