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Project Report by Mind Map: Project Report
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Project Report

knot theory

definitions

knot

knot projection

link

strands & under-/overcrossing

composite / prime

alternating

knot representations

planar, planar diagrams, Dowker codes / Gauss codes, planar graphs

3-D, knot complement, polygonal

closed braids

Conway, def: tangle

history & application

Gauss: initial

Kelvin: knotted vortices

tabulation, Tait, Hoste, Rankin

applications, biology: knotted DNA, chemistry: knotted molecules, physics

knot equivalence

diagrammatic moves, Reidemeister, Flype, Markov moves on braids

invariants, tricolourability, polynomials, others, vassiliev, (?) minimum braid

special case: unknotting, Haken, NP bound, (?) Birman, Test cases, Monster, Freedman, Ochiai

types of knots, torus, satellite, hyperbolic

heuristic search

introduction

origin

definitions, optimisation problem, neighbourhood, cost/fitness function

types, local search, idea, examples, simulated annealing, tabu search, genetic algorithms, idea

simulated annealing

idea: metallurgy

pseudocode

parameters, cooling schedule, acceptance test

properties, theoretical convergence

tabu search

idea

pseudocode

parameters, adaptive memory, aspiration criteria, replacement criteria

properties, customisability

genetic algorithms

pseudocode

parameters, fitness function, reproduction, cross-over, mutation, selection

properties, non-local, parallel

applications to knot theory

unknotting, 3-D, energy functions, Furihara, Buck & Simon, heuristic / gradient descent, planar, comparison to 3-D, examples, MING: Energy minimisation by gradient descent, Huang: Unknotting by annealing, Sharein: gradient descent, Ladd/Kavraki: Motion planning, Weaver: Planar annealing

other, Johnson: disproving conjectures

Plug & Play Tool

requirements

functional, flexible, extensible, efficient, manual / automated

process, timely / plannable, scalable, incremental

key functionalities, knot representation & manipulation, search, statistics & visualisation

architecture

process, comparison, waterfall, v-model, agile / xp, test-driven, incremental, (?) refactoring

technology, language, ruby + C, flexible, fast, shell interfacing + GUI, external tools, gnuplot

design

high-level structure, gui, experiment, load & save, heuristic search, problem, neighbourhood, cost, solution, knot manipulation

heuristic search, interface, criteria, simulated annealing, cooling schedule

knot manipulation, criteria, complete, unambiguous, manipulable, efficient, (?) scalable, (?) evaluable, availability of data, planar, data structures, interface, 3-D, data structures, interface

V & V

unit testing

integration testing

sanity checks

Experimentation

test cases

categories, torus/satellite/hyperbolic, prime / composite, reduced / non-reduced, mutant: KT, chiral, unknots: Monster, Freedman, Ochiai, knots: Perko

list of projections, caveats, generation, dowker codes, culling, pipeline, efficiency concerns

preliminary experiments

redo weaver

add ochiai

random perturbation very weak

different weights for Reid moves

planar direct search

testing for identical projections, graph isomorphism, cost functions, criteria, discrimination, guidance, degree of isomorphism, sum of characteristics, sum + grounding term, Weisfeiler Lehman cost

one sided / two sided

bounds, classifying projections, finding bounds, results, other moves: flypes, other identity test: only WL cost

planar statistical

initial: search results

search process statistics, sources, accept streak, cost function, problems, higher crossing numbers, -> use multiple sources, requires bounds, cannot manage chirality, -> use special cost function, other areas, word problem: braids, graph isomorphism

3-D

experimentation, same as for statistical

results, harder to find global optima, weird: deeper search -> worse results

Evaluation

planar direct search

proof of concept

bounds

statistical

automatic "invariants"

transferable

software engineering

Objectives

insufficiencies

only unknotting, uniqueness below threshold, unique global minima

local minima (?)

negative results

knot equivalence

bounds

statistical methods

tool

efficient exploration, multiple knot representations, multiple heuristic searches, multiple cost functions

Conclusion