/* Copyright (c) 1997-2021
   Ewgenij Gawrilow, Michael Joswig, and the polymake team
   Technische Universität Berlin, Germany
   https://polymake.org

   This program is free software; you can redistribute it and/or modify it
   under the terms of the GNU General Public License as published by the
   Free Software Foundation; either version 2, or (at your option) any
   later version: http://www.gnu.org/licenses/gpl.txt.

   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
--------------------------------------------------------------------------------
*/

#include "polymake/client.h"
#include "polymake/Array.h"
#include "polymake/Matrix.h"
#include "polymake/Set.h"
#include "polymake/topaz/complex_tools.h"
#include "polymake/topaz/hasse_diagram.h"
#include "polymake/Rational.h"
#include "polymake/list"

namespace polymake { namespace topaz {

BigObject unknot(const Int m, const Int n, OptionSet options)
{
   if (m < 2 || n < 1)
      throw std::runtime_error("unknot: m>=2 and n>=1 required\n");

   Rational eps;
   if (!(options["eps"] >> eps))
      eps = Rational(1, 200*(n+m+2));

   std::list<Set<Int>> C;
   const Int k1 = 2*(m+1)-1;
   const Int k2 = 2*(n+m+2)-1;
   const Int k3 = 2*(2*n+m+3)-1;
   const Int k4 = 2*(2*n+2*m+3)-1;
   const Int a = k4+1;

   // add "crossing" simplices
   Set<Int> f;
   f+=0; f+=1;
   for (Int i=3; i<=k1; i+=2) {
      f-=i-5; f-=i-4; f+=i-1; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=k1+1; f+=k1+2;
   for (Int i=k1+4; i<=k2; i+=2) {
      f-=i-5; f-=i-4; f+=i-1; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=k2+1; f+=k2+2;
   for (Int i=k2+4; i<=k3; i+=2) {
      f-=i-5; f-=i-4; f+=i-1; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=k3+1; f+=k3+2;
   for (Int i=k3+4; i<=k4; i+=2) {
      f-=i-5; f-=i-4; f+=i-1; f+=i;
      C.push_back(f);
   }

   // add "cone" simplices of the knot
   f.clear();
   f+=a; f+=0; f+=1;
   for (Int i=2; i<=k1; ++i) {
      f-=i-3; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=a; f+=k1+1; f+=k1+2;
   for (Int i=k1+3; i<=k2; ++i) {
      f-=i-3; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=a; f+=k2+1; f+=k2+2;
   for (Int i=k2+3; i<=k3; ++i) {
      f-=i-3; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=a; f+=k3+1; f+=k3+2;
   for (Int i=k3+3; i<=k4; ++i) {
      f-=i-3; f+=i;
      C.push_back(f);
   }

   // add fill in "cone" simplices
   f.clear();
   f+=a; f+=k1+1; f+=1;
   for (Int i=3; i<=k1+2; i+=2) {
      f-=i-4; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=a; f+=k2+1; f+=0;
   for (Int i=2; i<=k1-1; i+=2) {
      f-=i-4; f+=i;
      C.push_back(f);
   }
   f-=k1-3; f+=k2+2;
   C.push_back(f);

   f.clear();
   f+=a; f+=k2-1; f+=k4-1;
   for (Int i=k4-3; i>=k3+1; i-=2) {
      if (i+4!=a) f-=i+4;
      f+=i;
      C.push_back(f);
   }
   f-=k3+3; f+=k2;
   C.push_back(f);

   f.clear();
   f+=a; f+=k3-1; f+=k4;
   for (Int i=k4-2; i>=k3; i-=2) {
      f-=i+4; f+=i;
      C.push_back(f);
   }

   f.clear();
   f+=a; f+=k1-1; f+=k1;
   for (Int i=k1+2; i<=k2; i+=2) {
      f-=i-4; f+=i;
      C.push_back(f);
   }
   f-=k2-2; f+=k3+1;
   C.push_back(f);

   f.clear();
   f+=a; f+=k3+1; f+=k3+2;
   for (Int i=k3; i>=k2+2; i-=2) {
      f-=i+4; f+=i;
      C.push_back(f);
   }
   f-=k2+4; f+=k1-1;
   C.push_back(f);

   // add cone over boundary
   const Lattice<BasicDecoration> HD = hasse_diagram_from_facets(Array<Set<Int>>(C));
   const auto B = boundary_of_pseudo_manifold(HD);
   for (auto b = entire(B); !b.at_end(); ++b)
      C.push_back(b->face + scalar2set(a+1));

   // compute knot
   std::list<Set<Int>> K;

   for (Int i=3; i<=k1; i+=2) {
      Set<Int> e;
      e+=i-3; e+=i;
      K.push_back(e);
      e.clear();
      e+=i-2; e+=i-1;
      K.push_back(e);
   }

   for (Int i=k1+4; i<=k2; i+=2) {
      Set<Int> e;
      e+=i-3; e+=i;
      K.push_back(e);
      e.clear();
      e+=i-2; e+=i-1;
      K.push_back(e);
   }

   for (Int i=k2+4; i<=k3; i+=2) {
      Set<Int> e;
      e+=i-3; e+=i;
      K.push_back(e);
      e.clear();
      e+=i-2; e+=i-1;
      K.push_back(e);
   }

   for (Int i=k3+4; i<=k4; i+=2) {
      Set<Int> e;
      e+=i-3; e+=i;
      K.push_back(e);
      e.clear();
      e+=i-2; e+=i-1;
      K.push_back(e);
   }

   Set<Int> e;
   e+=k1; e+=k1+2;
   K.push_back(e);

   e.clear();
   e+=k1-1; e+=k2+2;
   K.push_back(e);

   e.clear();
   e+=k2; e+=k3+1;
   K.push_back(e);

   e.clear();
   e+=k3; e+=k3+2;
   K.push_back(e);

   e.clear();
   e+=1; e+=k1+1;
   K.push_back(e);

   e.clear();
   e+=0; e+=k2+1;
   K.push_back(e);

   e.clear();
   e+=k2-1; e+=k4-1;
   K.push_back(e);

   e.clear();
   e+=k3-1; e+=k4;
   K.push_back(e);

   // embedded as a subcomplex
   BigObject KSC("SimplicialComplex", "INPUT_FACES", K);

  // geometric realization
  Matrix<Rational> Coordinates(a+2,3);
  for (Int i = 0; i <= k1; ++i){
    Coordinates(i,0) = Rational(2-(i%2));
    Coordinates(i,1) = Rational(i/2);
    Coordinates(i,2) = i%2==0 ? -Rational(i/2)*eps  : Rational(i/2)*eps;
  }

  for (Int i = k1+1; i <= k2; ++i){
    Coordinates(i,0)=Rational(i%2);
    Coordinates(i,1)=Rational(i/2);
    Coordinates(i,2) = i%2==0 ? -Rational((i-k1-1)/2)*eps : Rational((i-k1-1)/2)*eps;
  }

  for (Int i = k2+1; i <= k3; ++i){
    Coordinates(i,0)=Rational(3-(i%2));
    Coordinates(i,1)=Rational((i-k2+k1)/2);
    Coordinates(i,2) = i%2==0 ? -Rational((i-k2-1)/2)*eps : Rational((i-k2-1)/2)*eps;
  }

  for (Int i = k3+1; i <= k4; ++i){
    Coordinates(i,0)=Rational(1+(i%2));
    Coordinates(i,1)=Rational((i-k3+k2)/2);
    Coordinates(i,2) = i%2==0 ? -Rational((i-k3-1)/2)*eps : Rational((i-k3-1)/2)*eps;
  }

  Coordinates(a, 0) = Rational(3,2);
  Coordinates(a, 1) = Rational((k1+k2)/4);
  Coordinates(a, 2) = Rational(-2);
  Coordinates(a+1, 0) = Rational(3,2);
  Coordinates(a+1, 1) = Rational((k1+k2)/4);
  Coordinates(a+1, 2) = Rational(2);

  BigObject p("GeometricSimplicialComplex<Rational>",
              "FACETS", C,
              "COORDINATES", Coordinates,
              "KNOT", KSC);
  p.set_description() << "Vicious embedding of the unknot in the 1-skeleton of the 3-sphere.\n";
  return p;
}

UserFunction4perl("# @category Producing from scratch\n"
                  "# Produces a triangulated 3-sphere with the particularly NASTY embedding\n"
                  "# of the unknot in its 1-skeleton. The parameters //m// >= 2 and //n// >= 1\n"
                  "# determine how entangled the unknot is. //eps// determines the [[COORDINATES]].\n"
                  "# @param Int m"
                  "# @param Int n"
                  "# @option Rational eps"
                  "# @return GeometricSimplicialComplex",
                  &unknot, "unknot($$ { eps => undef })");
} }

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