anagorko / markov_curves

Markov curves

Markov complexes

The new codebase is under development. Section "Diagram generator for Markov curves" below contains information about the earlier approach.

Installation

Create and activate a new virtual environment.

python3 -m venv ~/envs/markov_curves/
source ~/envs/markov_curves/bin/activate

Install dependencies

python3 -m pip install -r requirements.txt

Diagram generator for Markov curves (old approach)

This repository contains software that imlements diagram representations of Markov compacta as described in

G. C. Bell, A. Nagórko "Detecting topological properties of Markov compacta with combinatorial properties of their diagrams"

Creating new diagrams

Below we describe steps needed to extend elementary.py to generate diagrams and sequence for paper

G. C. Bell, A. Nagórko "A construction of Nobeling manifolds of arbitrary weight"

Steps:

• For each production, define a subclass of the Production class. Here we defined NobelingPoint and NobelingEdge productions (this is elementary Markov sequence).
• For each gluing, define a subclass of the Gluing class. Here we defined NobelingGluing_Left and NobelingGluing_Right for gluing NobelingPoint production over left and right end of NobelingEdge.
• Create subclass of ElementaryMarkovDiagram that contains information about the starting graph, productions and gluings.

Implementation Details

Productions

class NobelingPoint(Production):
	@classmethod
	def init(cls):
		cls.top = top = Graph()
		a, b, c = top.add_vertex(), top.add_vertex(), top.add_vertex()
		top.add_edge(a, b)
		top.add_edge(b, c)
		top.add_edge(a, c)
		pos = top.new_vertex_property("vector<double>")
		pos[a] = (-0,0)
		pos[b] = (1,0.2)
		pos[c] = (0.8, 0.6)
		top.vp["pos"] = pos

		cls.bottom = bot = Graph()
		A = bot.add_vertex()
		pos = bot.new_vertex_property("vector<double>")
		pos[A] = (0, 0)
		bot.vp["pos"] = pos

		cls.bonding_map = Map(bot, top)
		cls.bonding_map[a] = A
		cls.bonding_map[b] = A
		cls.bonding_map[c] = A
NobelingPoint.init()

class NobelingEdge(Production):
	@classmethod
	def init(cls):
		cls.top = top = Graph()
		a, b, c, d, e, f = top.add_vertex(), top.add_vertex(), top.add_vertex(), top.add_vertex(), top.add_vertex(), top.add_vertex()
		top.add_edge_list([(a,b), (b,c), (a,c), (d,e), (d,f), (e,f), (a,d), (a,e), (a,f), (b,d), (b,e), (b,f), (c,d), (c,e), (c,f)])

		pos = top.new_vertex_property("vector<double>")
		pos[a] = (-1,0)
		pos[b] = (0,0.2)
		pos[c] = (-0.2, 0.6)
		pos[d] = (1,0)
		pos[e] = (2,0.2)
		pos[f] = (1.8, 0.6)
		top.vp["pos"] = pos

		cls.bottom = bot = Graph()
		A,B = bot.add_vertex(), bot.add_vertex()
		pos = bot.new_vertex_property("vector<double>")
		pos[A] = (-1, 0)
		pos[B] = (1, 0)
		bot.vp["pos"] = pos

		cls.bonding_map = Map(bot, top)
		cls.bonding_map[a] = A
		cls.bonding_map[b] = A
		cls.bonding_map[c] = A
		cls.bonding_map[d] = B
		cls.bonding_map[e] = B
		cls.bonding_map[f] = B
NobelingEdge.init()

Gluings

class NobelingGluing_Left(Gluing):
	@classmethod
	def init(cls):
		g1 = NobelingPoint.get_top()
		g2 = NobelingEdge.get_top()

		cls.top_glue = tg = Map(g1, g2)
		tg[g1.get_vertices()[0]] = g2.get_vertices()[0]
		tg[g1.get_vertices()[1]] = g2.get_vertices()[1]
		tg[g1.get_vertices()[2]] = g2.get_vertices()[2]

		g1 = NobelingPoint.get_bottom()
		g2 = NobelingEdge.get_bottom()
		cls.bottom_glue = bg = Map(g1, g2)
		bg[g1.get_vertices()[0]] = g2.get_vertices()[0]
NobelingGluing_Left.init()

class NobelingGluing_Right(Gluing):
	@classmethod
	def init(cls):
		g1 = NobelingPoint.get_top()
		g2 = NobelingEdge.get_top()

		cls.top_glue = tg = Map(g1, g2)
		tg[g1.get_vertices()[0]] = g2.get_vertices()[3]
		tg[g1.get_vertices()[1]] = g2.get_vertices()[4]
		tg[g1.get_vertices()[2]] = g2.get_vertices()[5]

		g1 = NobelingPoint.get_bottom()
		g2 = NobelingEdge.get_bottom()
		cls.bottom_glue = bg = Map(g1, g2)
		bg[g1.get_vertices()[0]] = g2.get_vertices()[1]
NobelingGluing_Right.init()

Diagram class

class NobelingDiagram(ElementaryMarkovDiagram):
	@classmethod
	def init(cls):
		cls.starting_graph = g = Graph(directed=False)
		pos = g.new_vertex_property("vector<double>")
		a=g.add_vertex()
		pos[a] = (0,1)
		b=g.add_vertex()
		pos[b] = (0,-1)
		g.add_edge(a,b)
		g.vp["pos"] = pos

		cls.productions = {
			"V": NobelingPoint,
			"E": NobelingEdge
		}
		cls.gluings = {
		 	"L": NobelingGluing_Left, 
			"R": NobelingGluing_Right
		}
NobelingDiagram.init()

Installation

Run make to generate diagrams.

Software dependencies:

• graph-tool
• python-termcolor