Drawing a coloured Karnaugh map on a torus

Question

I am trying to draw this diagram

This is where I've got to, but I'm stuck trying to work out how add the colours.

MWE

\documentclass{article}
\usepackage{karnaugh-map}
\usepackage{pgfplots}
\pgfplotsset{compat=1.14}

\begin{document}

\begin{tikzpicture}
    \begin{axis}
    [hide axis,
    plot box ratio = 1 6 1,
    view = {0}{15}]
       \addplot3[surf,
       samples=20,
       domain=0:2*pi,y domain=0:2*pi,
       z buffer=sort]
       ({(2+cos(deg(x)))*cos(deg(y+pi/2))}, 
        {(2+cos(deg(x)))*sin(deg(y+pi/2))}, 
        {sin(deg(x))});
    \end{axis}
\end{tikzpicture}
%
\begin{karnaugh-map}[4][4][1][][]
\end{karnaugh-map}

\end{document}

Colour codes

0000 - #F70400

0100 - #AA0154

1100 - #5600AB

1000 - #0003FB

0001 - #FF5500

0101 - #AA5455

1101 - #5555AB

1001 - #0055FE

0011 - #FFAA01

0111 - #AAA956

1111 - #56AAAA

1011 - #00AAFF

0010 - #FEFF02

0110 - #A9FF54

1110 - #55FFAA

1010 - #0FF6FF

Answer 1

Thanks for the colors! I had no time to adjust the details.

\documentclass{article}
\usepackage[rgb]{xcolor}
\usepackage{karnaugh-map}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\definecolor{mycolor0000}{HTML}{F70400}
\definecolor{mycolor0100}{HTML}{AA0154}
\definecolor{mycolor1100}{HTML}{5600AB}
\definecolor{mycolor1000}{HTML}{0003FB}
\definecolor{mycolor0001}{HTML}{FF5500}
\definecolor{mycolor0101}{HTML}{AA5455}
\definecolor{mycolor1101}{HTML}{5555AB}
\definecolor{mycolor1001}{HTML}{0055FE}
\definecolor{mycolor0011}{HTML}{FFAA01}
\definecolor{mycolor0111}{HTML}{AAA956}
\definecolor{mycolor1111}{HTML}{56AAAA}
\definecolor{mycolor1011}{HTML}{00AAFF}
\definecolor{mycolor0010}{HTML}{FEFF02}
\definecolor{mycolor0110}{HTML}{A9FF54}
\definecolor{mycolor1110}{HTML}{55FFAA}
\definecolor{mycolor1010}{HTML}{0FF6FF}


\pgfplotsset{colormap={BR}{%
color(0)=(mycolor0000) color(1)=(mycolor0100) color(2)=(mycolor1100) color(3)=(mycolor1000)
color(4)=(mycolor0001) color(5)=(mycolor0101) color(6)=(mycolor1101) color(7)=(mycolor1001)
color(8)=(mycolor0011) color(9)=(mycolor0111) color(10)=(mycolor1111) color(11)=(mycolor1011)
color(12)=(mycolor0010) color(13)=(mycolor0110) color(14)=(mycolor1110) color(15)=(mycolor1010)
}}
\begin{document}

\begin{tikzpicture}[font=\small\sffamily]
    \begin{axis}
    [hide axis,shader=flat corner,%colormap name=BR,
    plot box ratio = 1 6 1,
    view = {0}{15}]
       \addplot3[surf,
       samples=32,point meta={int(mod(-atan2(y,x)+45+360,360)/90)+
       4*int(mod(atan2(z,sqrt(x^2+y^2)-2)+360+180,360)/90)
       },domain=0:360,y domain=0:360,
       z buffer=sort]
       ({(2+cos(x))*cos(y+90)}, 
        {(2+cos(x))*sin(y+90)}, 
        {sin(x)});
    \node at ({(2+cos(45))*cos(-90)},{(2+cos(45))*sin(-90)},{cos(45)}) {0111};  
    \node at ({(2+cos(45))*cos(-90)},{(2+cos(45))*sin(-90)},{0.15-cos(45)}) {0101}; 
    \fill ({(2-cos(45))*cos(90-50)},{(2-cos(45))*sin(90-50)},{cos(80)}) circle (1mm);   
    \end{axis}
\end{tikzpicture}
%
\begin{karnaugh-map}[4][4][1][][]
\end{karnaugh-map}
\end{document}