Reproduce bracket polynomials

Question

I want to reproduce the following bracket polynomials, but I'm quite clueless on how to do so. From what I've searched, people often use tikz to reproduce these knots. But how to draw knot using tikz? Also, how to draw the Reidemeister moves (the diagram inside the circles in the below pictures) with tikz? Helps would be really appreciated.

Answer 1

I do not know what "4th and 5th picture" precisely refers to, but I hope they are included in the following collection, which makes use of the knots library as well as some trick (rubout) mentioned in its manual. One could make the code shorter by introducing a macro for \vcenter{\hbox{...}} (I once got downvoted for doing that...) and loading the braket package and so on, but these are details.

\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}