{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Antes de empezar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Gracias por la invitación."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Trabajo incipiente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Motivación, resultados previos e ideas para seguir."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Problema\n",
    "\n",
    "Dada una curva algebraica plana $\\mathcal{C}$, estudiar la topología del par $(\\mathbb{C}^2,\\mathcal{C})$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "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 $\\pi_1(\\mathbb{C}^2\\setminus C_2)$ 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": [
    "## SIROCCO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Biblioteca en C++ para calcular trenzas sobre segmentos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Cálculo numérico, pero certificado mediante aritmética intervalar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Integrada en SageMath como paquete opcional."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\langle x_{0} \\mid \\rangle</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\langle x_{0} \\mid \\rangle\n",
       "\\end{math}"
      ],
      "text/plain": [
       "Finitely presented group < x0 |  >"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(C.fundamental_group())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## SIROCCO: demostración"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\langle x_{0}, x_{1}, x_{3} \\mid x_{1}^{-1}\\cdot x_{0}\\cdot x_{1}\\cdot x_{0}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1} , x_{3}\\cdot x_{0}\\cdot x_{3}^{-1}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{0} , x_{1}^{-1}\\cdot x_{0}\\cdot x_{1}\\cdot x_{3}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1}\\cdot x_{1}\\cdot x_{3}^{-1}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1}\\cdot x_{1}\\cdot x_{3} , x_{3}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{1}\\cdot x_{0}^{-1}\\cdot x_{3}\\cdot x_{0}\\cdot x_{1}\\cdot x_{3}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{1}^{-1}\\rangle</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\langle x_{0}, x_{1}, x_{3} \\mid x_{1}^{-1}\\cdot x_{0}\\cdot x_{1}\\cdot x_{0}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1} , x_{3}\\cdot x_{0}\\cdot x_{3}^{-1}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{0} , x_{1}^{-1}\\cdot x_{0}\\cdot x_{1}\\cdot x_{3}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1}\\cdot x_{1}\\cdot x_{3}^{-1}\\cdot x_{1}^{-1}\\cdot x_{0}^{-1}\\cdot x_{1}\\cdot x_{3} , x_{3}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{1}\\cdot x_{0}^{-1}\\cdot x_{3}\\cdot x_{0}\\cdot x_{1}\\cdot x_{3}\\cdot x_{0}^{-1}\\cdot x_{3}^{-1}\\cdot x_{1}^{-1}\\rangle\n",
       "\\end{math}"
      ],
      "text/plain": [
       "Finitely presented group < x0, x1, x3 | x1^-1*x0*x1*x0*x1^-1*x0^-1, x3*x0*x3^-1*x0^-1*x3^-1*x0, x1^-1*x0*x1*x3*x1^-1*x0^-1*x1*x3^-1*x1^-1*x0^-1*x1*x3, x3*x0^-1*x3^-1*x1*x0^-1*x3*x0*x1*x3*x0^-1*x3^-1*x1^-1 >"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": [
    "- 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; margin-left: auto; margin-right: auto;\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Propiedades de los diagramas de Rudolf"
   ]
  },
  {
   "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": [
    "- Tienen al discriminante como puntos límite"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "- Localmente vienen dados por ceros de funciones holomorfas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Casos degenerados"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Considerar $y\\cdot(x+y)\\cdot(x-y)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Para cada $x$, $y\\in\\{0,x,-x\\}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Si dos de estos valores tienen la misma parte real, los tres la tienen."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Luego el diagrama de Rudolf tiene tres segmentos superpuestos (incluso después de cambio de proyección)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Casos no degenerados"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Con un cambio de proyección, se pueden suavizar los cruces del mismo par fuera del discriminante:\n",
    "\n",
    "![](suavizado.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "## Casos no degenerados\n",
    "\n",
    "Cruces consecutivos\n",
    "\n",
    "![](cruceconsecutivo.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Propiedades de los diagramas de Rudolf no degenerados"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Unión de segmentos lisos que van del infinito al discriminante."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Cruces triples consecutivos, o simples no consecutivos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- La trenza correspondiente a un camino cerrado:\n",
    "  - Si no rodea puntos del discriminante, es trivial\n",
    "  - Si rodea a un punto del discriminante, es la trenza local de la singularidad (determina Puiseux)\n",
    "  - Modificar el camino para atravesar un cruce, da un movimiento de Tietze en la trenza."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Movimientos de Tietze\n",
    "\n",
    "![](tietze.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "<img src=\"laberinto0.png\" style=\"max-height: 600px; margin-left: auto; margin-right:auto;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Diagramas de Rudolf\n",
    "\n",
    "<img src=\"laberinto1.svg\" style=\"max-height: 600px; margin-left:auto; margin-right:auto;\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Muchas gracias"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": []
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