{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Diagramas de Rudolf de curvas planas\n",
    "\n",
    "<p style=\"text-align:center\">Seminario Rubio de Francia</p>\n",
    "\n",
    "<p style=\"text-align:center\">Miguel Angel Marco Buzunariz</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Geometría algebraica\n",
    "\n",
    "Considerar un sistema de ecuaciones polinómicas\n",
    "\n",
    "$$f_1(x_1,\\ldots,x_n)$$\n",
    "$$f_2(x_1,\\ldots,x_n)$$\n",
    "$$\\vdots$$\n",
    "$$f_m(x_1,\\ldots,x_m)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "El conjunto $V$ de sus soluciones forman una *variedad algebraica*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Problema general de la geometría algebraica\n",
    "\n",
    "Estudiar la geometría de $V$ usando la estructura algebraica de las ecuaciones que la definen."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Nos vamos a restringir a los casos más fáciles:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $m=1, n=2$ (curvas planas)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $\\mathbb{K}=\\mathbb{C}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Resultados clásicos conocidos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Si la curva es lisa: teoría de las superficies de Riemann\n",
    "  - Superficies de género $g=\\frac{d(d-1)}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ejemplo \n",
    "\n",
    "$y^2= x^2-1$\n",
    "\n",
    "![](hiperbola.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "La proyección sobre el eje $x$ es $2:1$ excepto en $x=\\pm 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ejemplo\n",
    "\n",
    "![](supriemannelev-0.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ejemplo\n",
    "\n",
    "<video data-autoplay src=\"supriemannelev.mp4\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Topología del encaje\n",
    "\n",
    "$C\\subseteq \\mathbb{C}^2$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Codimensión 2: situación análoga a los nudos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Uno de los invariantes principales es el grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "![](monodtrenzas1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "![](monodtrenzas2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "![](monodtrenzas3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas4.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas5.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas6.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas7.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas8.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Lema</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "    Esta construcción determina un morfismo de grupos $\\pi_1(\\mathbb{C}\\setminus \\Delta)\\longrightarrow B_d$.\n",
    "    </p>\n",
    "</div>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Definición</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "    Este morfismo recibe el nombre de <i>monodromía de trenzas</i> de la curva.\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Se puede representar como una lista de trenzas correspondientes a una base geométrica de $\\pi_1(\\mathbb{C}\\setminus \\Delta)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas9.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<video data-autoplay src=\"monodtrenzas10.webm\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Monodromía de trenzas\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Teorema</b> (Zariski-Van Kampen)\n",
    "    </h3>\n",
    "    <p>\n",
    "    El grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$ es el cociente del grupo libre $F_d$ por la acción de la monodromía de trenzas.\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Teorema</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "    La monodromía de trenzas determina completamente la topología del par $(\\mathbb{C}^2,C)$.\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Pares de Zariski\n",
    "\n",
    "La monodromía de trenzas es un objeto algebraico (morfismo de grupos) definido a partir de otro objeto algebraico (ideal en el anillo de polinomios)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Definición</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "        Dos curvas algebraicas $C_1$ y $C_2$ forman un <i>par de Zariski aritmético</i> si\n",
    "    <ul>\n",
    "        <li class=\"fragment\">Los polinomios que las definen $f_1,f_2$ son conjugados de Galois.</li>\n",
    "        <li class=\"fragment\">Los grupos fundamentales $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ y $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ son no isomorfos.</li>\n",
    "    </ul>\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Pares de Zariski\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Teorema</b> (ACG_)\n",
    "    </h3>\n",
    "    <p>\n",
    "        Existen pares de Zariski aritméticos\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Esto implica que la monodromía de trenzas (o el grupo fundamental) no se puede calcular de manera puramente algebraica."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Se necesitan cálculos **numéricos**, pero **certificados**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo de la monodromía de trenzas\n",
    "\n",
    "Primer paso: elegir los caminos\n",
    "\n",
    "![](voronoi0.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo de la monodromía de trenzas\n",
    "\n",
    "Primer paso: elegir los caminos\n",
    "\n",
    "![](voronoi1.jpg)\n",
    "\n",
    "El *Diagrama de Voronoi* nos da segmentos que rodean los puntos del discriminante, acercándose lo menos posible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo de la trenza sobre un segmento\n",
    "\n",
    "Mediante cambio de variable, podemos normalizar a $[0,1]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "![](trenzasola.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo de la trenza sobre un segmento\n",
    "\n",
    "Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
    "\n",
    "![](trenzapoligonal.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo de la trenza sobre un segmento\n",
    "\n",
    "Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
    "\n",
    "![](trenzaentorno.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Aritmética intervalar\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Definición</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "        Un <i>intervalo complejo</i> es un conjunto $$[r_0,r_1]+[i_0,i_1]\\cdot i:=\\{a+b\\cdot i \\mid a \\in [r_0,r_1], b\\in [i_0,i_1]\\}\\subseteq \\mathbb{C}$$\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Definición</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "        Dado $Y\\subseteq \\mathbb{C}$, su <i>clausura intervalar</i> $[Y]$ es el menor intervalo que lo contiene.\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Se puede operar con intervalos tomando la clausura intervalar de las operaciones punto a punto."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Los intervalos con estas operaciones **no** forman un anillo (pero para ciertos efectos, se puede operar como si lo fuera)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Criterio del intervalo de Newton\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Teorema</b>\n",
    "    </h3>\n",
    "    <p>\n",
    "        Sea $Y$ un intervalo complejo, $y0\\in Y$. Sea $f: Y\\longrightarrow \\mathbb{C}$ una función holomorfa, tal que $0\\in [f'(Y)]$. Si\n",
    "        $$\n",
    "        N(f,y_0,Y):=y_0-\\frac{f(y_0)}{[f'(Y)]}\\subseteq Y\n",
    "        $$\n",
    "        entonces $f$ tiene un único cero en $Y$. Además, este cero está en $N(f,y_0,Y)$.\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Podemos usar este teorema para asegurar que tenemos un entorno alrededor de una raíz aproximada, que contiene a la raíz auténtica, pero sin tocar a las demás raíces."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Notar que, ya que estamos operando con intervalos, podemos usar intervalos en los propios coeficientes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ejemplo\n",
    "\n",
    "$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
    "$$y_0=1$$\n",
    "$$Y=[0.5,1.5]$$\n",
    "$$x_0=0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$=1-\\frac{[-0.1,0.1]}{[0.9,3.3]}\\approx [0.888,1.111]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ejemplo: extender $x$ a un intervalo\n",
    "\n",
    "$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
    "$$y_0=1$$\n",
    "$$Y=[0.5,1.5]$$\n",
    "$$x=[0,0.25]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1-[0,0.25]}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$=1-\\frac{[-0.35,0.1]}{[0.9,3.3]}\\approx [0.888,1.13888]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos4.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos5.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Cálculo numérico certificado de la trenza\n",
    "\n",
    "![](cubos6.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## SIROCCO: demostración"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A.<x,y> = AffineSpace(QQ,2)\n",
    "C = A.curve(y^2-x^3-x^2)\n",
    "C.plot((x,-3,3),(y,-3,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "show(C.fundamental_group())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## SIROCCO: demostración"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "C2 = A.curve((x^2+y^2)^2+18*(x^2+y^2) - 27 -8*(x^3-3*x*y^2))\n",
    "C2.plot((x,-3,3),(y,-3,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "show(C2.fundamental_group())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "Volviendo a las superficies de Riemann:\n",
    "\n",
    "![](diagramarudolfhiperbola2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "Volviendo a las superficies de Riemann:\n",
    "\n",
    "![](diagramarudolfhiperbola1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "<div class=\"thm\">\n",
    "    <h3>\n",
    "        <b>Definición</b> \n",
    "    </h3>\n",
    "    <p>\n",
    "        Sea $f\\in\\mathbb{C}[x,y]$, su <i>diagrama de Rudolf</i> es el conjunto\n",
    "        $$\\{x\\in\\mathbb{C} \\mid \\exists y_0\\neq y_1, Re(y_0)=Re(y_1), f(x,y_0)=f(x,y_1)=0\\}$$\n",
    "    </p>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Como conjunto de $\\mathbb{R}^2\\cong \\mathbb{C}$, es un conjunto semialgebraico"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- En general, son difíciles de calcular"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "Diagrama de Rudolf de $(4z^3+12z+3-w^2)^2-8w-48z+52$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<img src=\"diagramarudolforevkov.svg\" style=\"max-height:450px !important; height:450px;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "![](laberinto0.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "![](laberinto1.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Muchas gracias"
   ]
  }
 ],
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