重现括号多项式

Question

我想重现以下括号多项式，但我对如何做到这一点一无所知。从我搜索的内容来看，人们经常使用 tikz 来重现这些结。但如何使用 tikz 绘制结？此外，如何使用 tikz 绘制 Reidemeister 动作（下图中圆圈内的图表）？非常感谢您的帮助。

Answer 1

我不知道“第 4 和第 5 张图片”具体指什么，但我希望它们包含在以下集合中，该集合利用了库knots以及rubout其手册中提到的一些技巧 ()。可以通过引入宏\vcenter{\hbox{...}}（我曾经因为这样做而被否决......）和加载braket包等来缩短代码，但这些都是细节。

\documentclass[fleqn]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{% this is all from https://loopspace.mathforge.org/HowDidIDoThat/TeX/Knots/
  knots,
  hobby,
  decorations.pathreplacing,
  shapes.geometric,
  calc
}

\tikzset{mypoly/.style={ultra thick}}

\tikzset{% and this is from Andrew Stacey's homepage except that I removed "red";-)
  knot diagram/every strand/.append style={
    mypoly
  },
  show curve controls/.style={
    postaction=decorate,
    decoration={show path construction,
      curveto code={
        \draw [blue, dashed]
        (\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
        node [at end, draw, solid, inner sep=2pt]{};
        \draw [blue, dashed]
        (\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
        node [at start, draw, solid, inner sep=2pt]{}
        node [at end, fill, blue, ellipse, inner sep=2pt]{}
        ;
      }
    }
  },
  show curve endpoints/.style={
    postaction=decorate,
    decoration={show path construction,
      curveto code={
        \node [fill, blue, ellipse, inner sep=2pt] at (\tikzinputsegmentlast) {}
        ;
      }
    }
  }
}
\begin{document}
\begin{align}
\left\langle\vcenter{\hbox{
\begin{tikzpicture}
\begin{knot}[
  consider self intersections=true,
  clip width=3,
  flip crossing=1,
  ignore endpoint intersections=false
  ]
\strand[overlay] (-1,0) to[out=90,in=135] (0,0) to[out=-45,in=-90]  (1,0)
to[out=90,in=45] (0,0) to[out=-135,in=-90] cycle;
\path[use as bounding box] (-1.05,-0.35) rectangle (1.05,0.35);
\end{knot}
\end{tikzpicture}}}\right\rangle&=
q\,\left\langle\vcenter{\hbox{\begin{tikzpicture}
\draw[mypoly] (-1,0) to[out=90,in=160] (-0.8,0.3) 
to[out=-20,in=90] (-0.1,0) to[out=-90,in=20] (-0.8,-0.3) to[out=-160,in=-90] cycle;
\draw[mypoly] (1,0) to[out=90,in=20]  (0.8,0.3) 
 to[out=-160,in=90] (0.1,0) to[out=-90,in=160] (0.8,-0.3) to[out=-20,in=-90] cycle;
\end{tikzpicture}}}\right\rangle
+q^{-1}\,
\left\langle\vcenter{\hbox{\begin{tikzpicture}
\draw[mypoly] (-1,0) to[out=90,in=160] (-0.8,0.3) 
to[out=-20,in=180] (0,0.15) to[out=0,in=-160] (0.8,0.3) to[out=20,in=90] (1,0)
to[out=-90,in=-20] (0.8,-0.3) to[out=160,in=0] (0,-0.15) to[out=180,in=20] 
(-0.8,-0.3) to[out=-160,in=-90] cycle;
\end{tikzpicture}}}\right\rangle\notag\\
&=q\,\delta^2+q^{-1}\,\delta=-q^3\,\delta\;.
\end{align}
\dots
\begin{align}
\left\langle\vcenter{\hbox{\begin{tikzpicture}
\begin{knot}[
  consider self intersections=true,
  clip width=3,
  flip crossing=1,
  ignore endpoint intersections=false
  ]
\strand[overlay,xscale=-1] (-1,0) to[out=90,in=160] (-0.8,0.3) 
to[out=-20,in=180] (0,0.15) to[out=0,in=-160] (0.8,0.3) to[out=20,in=90] (1,0)
to[out=-90,in=-20] (0.8,-0.3) to[out=160,in=-90] (-0.2,-0.2)
to[out=90,in=90] (0.2,-0.2) to[out=-90,in=20]
(-0.8,-0.3) to[out=-160,in=-90] cycle;
\path[use as bounding box] (-1.05,-0.35) rectangle (1.05,0.35);
\end{knot}
\end{tikzpicture}}}\right\rangle=-q^3\,\delta\;.
\end{align}

\tikzset{rubout/.style={ultra thick,double=black,draw=gray!30,double distance=1.6pt}}

\begin{align}
q^4\,J\!\left(\vcenter{\hbox{\begin{tikzpicture}
 \node[circle,fill=gray!30,minimum width=1cm] (circ){};
 \draw[mypoly] (circ.-45) -- (circ.135);
 \draw[rubout] (circ.45) -- (circ.-135);
\end{tikzpicture}}}\right)+
q^{-4}\,J\!\left(\vcenter{\hbox{\begin{tikzpicture}
 \node[circle,fill=gray!30,minimum width=1cm] (circ){};
 \draw[mypoly] (circ.45) -- (circ.-135);
 \draw[rubout] (circ.-45) -- (circ.135);
\end{tikzpicture}}}\right)&=\left(q^2+q^{-2}\right)\,J\!\left(
\vcenter{\hbox{\begin{tikzpicture}
 \node[circle,fill=gray!30,minimum width=1cm] (circ){};
 \draw[mypoly] (circ.45) to[out=-135,in=135] (circ.-45);
 \draw[mypoly] (circ.135) to[out=-45,in=45] (circ.-135);
\end{tikzpicture}}}\right)
\end{align}
\end{document}