On the derivation of the HOMFLYPT polynomial invariant for fluid knots
On the derivation of the HOMFLYPT polynomial invariant for fluid knots
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DOI：
10.1017/jfm.2015.231
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发表时间：
2015-05
影响因子：
3.7
通讯作者：
Xin Liu;Renzo L. Ricca
中科院分区：
文献类型：
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作者：
Xin Liu;Renzo L. Ricca
通过使用和推广先前的结果(Liu&Ricca，J.Phys.A，Vol.45,2012,205501)，我们从螺旋度出发导出了理想流体纽结的HOMFLYPT多项式的斜率关系，从而严格地证明了HOMFLYPT多项式是拓扑流体力学的一个新的、强大的不变量。由于这个不变量是一个两个变量的多项式，所以斜率关系是从两个独立的方程推导出来的，这些方程用扭度和扭度贡献表示。扭动是通过对假想的局部路径进行加/减，而扭曲是通过德恩的手术。然后，HOMFLYPT成为节点拓扑和场强的函数。为了说明这一点，我们推导出了一些基本情形的显式表达式，并将这些结果应用于均匀涡旋纠缠。通过检验一些特殊的例子，我们展示了HOMFLYPT多项式的数值实现如何提供对真实流体流动的流体力学行为的新见解。
By using and extending earlier results (Liu & Ricca, J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.