In this work, a general Monte Carlo framework is proposed for applying numerical knot invariants in simulations of systems containing knotted one-dimensional ring-shaped objects like polymers and vortex lines in fluids, superfluids or other quantum liquids. A general prescription for smoothing the sharp corners appearing in discrete knots consisting of segments joined together is provided. Smoothing is very important for the correct evaluation of numerical knot invariants.A discrete version of framing is adopted in order to eliminate singularities that are possibly arising when computing the invariants. The presented algorithms for smoothing, eliminating potentially dangerous singularities and speeding up the calculations are quite general and can be applied to any discrete knot defined off- or on-lattice.This is one of the first attempts to use numerical knot invariants in order to avoid potential topology breakings during the sampling process taking place in computer simulations, in which millions of knot conformations are randomly generated. As an application, the energy domain of knotted polymer rings subjected to short-range interactions is studied using the so-called Vassiliev knot invariant of degree 2.

中文翻译：

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数值拓扑不变量在打结环模拟中的应用：一种综合的蒙特卡罗方法

在这项工作中，提出了一个通用的 Monte Carlo 框架，用于在模拟包含打结的一维环状物体（如流体、超流体或其他量子液体中的聚合物和涡线）的系统中应用数值结不变量。提供了一种用于平滑出现在由连接在一起的段组成的离散结中的尖角的一般方法。平滑对于数值结不变量的正确评估非常重要。为了消除计算不变量时可能出现的奇异性，采用了离散版本的框架。所提出的用于平滑、消除潜在危险的奇异点和加速计算的算法非常通用，可以应用于任何定义的非晶格或晶格上的离散结。这是使用数值结不变量的第一次尝试，以避免在计算机模拟中发生的采样过程中潜在的拓扑破坏，其中数百万个结构象是随机生成的。作为一种应用，使用所谓的 2 次 Vasiliev 结不变量来研究经受短程相互作用的打结聚合物环的能量域。