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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="">
<meta name="author" content="">
<link rel="shortcut icon" href="assets/ico/favicon.ico">
<title>Torus Knot Fibration Visualization</title>
<!-- Bootstrap core CSS -->
<link href="assets/css/bootstrap.css" rel="stylesheet">
<!-- Custom styles for this template -->
<link href="assets/css/style.css" rel="stylesheet">
<link href="assets/css/font-awesome.min.css" rel="stylesheet">
<script src="assets/js/jquery.min.js"></script>
<!-- <script src="assets/js/modernizr.js"></script>-->
<!-- mathbox -->
<script src="./lib/mathbox/build/mathbox-bundle.js"></script>
<link rel="stylesheet" href="./lib/mathbox/build/mathbox.css">
<meta name="viewport" content="initial-scale=1, maximum-scale=1">
<!--math-->
<script src="./lib/math.min.js"></script>
<!--jquery-->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<!--bootstrap-->
<link rel="stylesheet" src="./lib/bootstrap-3.3.7-dist/css/bootstrap.min.css">
<link rel="stylesheet" src="./lib/bootstrap-3.3.7-dist/css/bootstrap-theme.min.css">
<script src="./lib/bootstrap-3.3.7-dist/js/bootstrap.min.js"></script>
<!--
bootstrap toggle
<link href="https://gitcdn.github.io/bootstrap-toggle/2.2.2/css/bootstrap-toggle.min.css" rel="stylesheet">
<script src="https://gitcdn.github.io/bootstrap-toggle/2.2.2/js/bootstrap-toggle.min.js"></script>
-->
<link href="./lib/switch/css/bootstrap3/bootstrap-switch.min.css" rel="stylesheet">
<script src="./lib/switch/js/bootstrap-switch.min.js"></script>
<!--bootstrap slider-->
<link href="./lib/slider/bootstrap-slider.min.css" rel="stylesheet">
<script src="./lib/slider/bootstrap-slider.min.js"></script>
<!-- my css -->
<link href="./lib/css/mystyle.css" rel="stylesheet">
<!-- mathjax-->
<script type="text/javascript" async src="assets/MathJax-2.6/MathJax.js?config=TeX-MML-AM_CHTML">
</script>
</head>
<body>
<!-- Fixed navbar -->
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">Torus Knot Fibration Visualization</a>
</div>
<div class="navbar-collapse collapse navbar-right">
<ul class="nav navbar-nav">
<li class="active"> <a href=index.html>HOME</a></li>
<li><a href="torusknot.html">TORUS & KNOT</a></li>
<li><a href="boundary.html">BOUNDARY</a></li>
<li><a href="surface.html">SURFACE</a></li>
<li><a href="stereo.html">STEREOGRAPHIC</a></li>
</ul>
</div>
<!--/.nav-collapse -->
</div>
</div>
<!-- *****************************************************************************************************************
BLUE WRAP
***************************************************************************************************************** -->
<div id="blue">
<div class="container">
<div class="row">
<h3>Introducing Knot Fibration</h3>
</div>
<!-- /row -->
</div>
<!-- /container -->
</div>
<!-- /blue -->
<!-- *****************************************************************************************************************
VISUALIZER
***************************************************************************************************************** -->
<div class="container-fluid mtb">
<div class="row">
<div id="torusvis" class="torusvis col-lg-7">
<script type="text/javascript" language="javascript">
console.log($('.torusvis'));
$('.torusvis').css('height', 0.8 * $(window).height() + 'px');
</script>
<script>
var mathbox = mathBox({
plugins: ['core', 'controls', 'cursor'],
controls: {
// Orbit controls, i.e. Euler angles, with gimbal lock
klass: THREE.OrbitControls,
parameters: {
noZoom: true
},
// Trackball controls, i.e. Free quaternion rotation
//klass: THREE.TrackballControls,
},
element: document.getElementById("torusvis"),
mathbox: {
warmup: 1
}
});
if (mathbox.fallback) throw "WebGL not supported"
var three = mathbox.three;
three.renderer.setClearColor(new THREE.Color('#f8f8f8'), 1.0);
var time = three.Time.clock;
// Place camera
var camera =
mathbox
.camera({
proxy: true,
position: [0, 2, 0],
lookAt: [0, 0, 0]
});
// 2D cartesian
var view =
mathbox
// .cartesian({
// position: [0, 0, -0.33],
// range: [
// [-math.pi, math.pi],
// [-math.pi, math.pi],
// [-math.pi, math.pi]
// ],
// scale: [1, 1, 1],
// });
.cartesian({
position: [0, 0, 0],
range: [
[-math.pi, math.pi],
[-math.pi, math.pi],
[-1, 1]
],
scale: [1, 1, 0.5],
});
function mod2pi(x) {
return (x % (2 * pi) + 2 * pi) % (2 * pi)
}
function linecut(x) {
var X = x
var thresh = 0.002;
if (Math.abs(pi) - X < thresh) {
X = NaN
}
return X
}
function fakemodpi(x) {
var tpi = 2 * pi;
var X = x;
//To stop edge errors:
while (Math.abs(X) > pi) {
if (X <= -pi) {
X = X + 2 * pi
} else if (X >= -pi + tpi) {
X = X - tpi
}
}
return linecut(X)
}
var alpha = 2;
//Quality of Life
var pi = math.pi;
var sin = math.sin;
var cos = math.cos;
// Torus Knot Parameters
var p = 2;
var q = 3;
var m = p / q;
var m2 = q / p;
var eps = 0.01;
function SurfaceCut(x) {
if (x <= 0) {
x = NaN;
} else if (x >= 2 * pi) {
x = NaN;
}
return x
}
function ZW(gamma, alpha, beta) {
var z = polarComplex(sin(gamma), alpha);
var w = polarComplex(cos(gamma), beta);
return [z, w];
};
function ReIm(z, w) {
return [z.re, z.im, w.re, w.im];
};
function polarComplex(r, phi) {
return math.complex({
r: r,
phi: phi
});
}
function sigma(z, w) {
return [w, z];
}
function SterProj(x, y, u, v) {
return [x / (1 - v), y / (1 - v), u / (1 - v)];
};
//First Piecewise Curve
function piecewise(x, y, t, n) {
x = fakemodpi(x);
var period = (pi) / q;
var amplitude = pi / (2 * p);
var wave = Math.pow(math.abs(4 / (pi) * y), 4) * ((2 * amplitude) / pi) * math.asin(sin((x + t / q) * pi / period)) + 2 * pi * n / p - (1 / p) * t
return wave
}
function SurfaceCut2(x) {
var cuteps = 0.2;
if ((x < 0) && (x > (0 - cuteps))) {
x = 0;
} else if ((x > 2 * pi) && (x < (2 * pi + cuteps))) {
x = 2 * pi;
} else if (x < (0 - cuteps)) {
x = NaN;
} else if (x > (2 * pi + cuteps)) {
x = NaN;
}
return x
}
console.log(piecewise(0, pi / 2, 0, 0))
view.area({
id: 'outsidearea',
expr: function(emit, x, y, i, j, t) {
tau = t % (2 * pi);
// var xdom = SurfaceCut(x);
var xdom = SurfaceCut2(x);
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,-1))));
emit(...SterProj(...ReIm(...ZW(y, xdom, piecewise(xdom, y, tau, 0)))));
emit(...SterProj(...ReIm(...ZW(y, xdom, piecewise(xdom, y, tau, 1)))));
// emit(...ReIm(...ZW(y,xdom,piecewise(x,y,tau,2))));
},
rangeX: [0, 3 * pi],
rangeY: [0, pi / 4],
width: 200,
height: 50,
channels: 3,
items: 2
});
view.surface({
id: 'outsidesurface',
color: '#384452',
proximity: 10000000000,
shaded: false,
crossed: false,
lineX: false,
visible: true,
zBias: 10
});
view.line({
id: 'outsidemesh',
color: '#00B3F3',
zBias: 10,
opacity: 0.4,
visible: true,
})
//Second Stereographic Surface
function piecewise2(x, y, t, n) {
x = x % (2 * pi);
var period = (pi) / p;
var amplitude = pi / (2 * q);
var wave = Math.pow(math.abs(4 / pi * y), 4) * ((2 * amplitude) / pi) * math.asin(sin((x - t) * pi / period)) - 2 * pi * n / q
return -(wave - pi / q + (1 / q) * t)
}
view.area({
id: 'insidearea',
expr: function(emit, x, y, i, j, t) {
var tau = t % (2 * pi);
// var xdom = SurfaceCut(x);
var xdom = -(x - (1 / p) * tau) - pi / p;
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 0), xdom))));
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 1), xdom))));
emit(...SterProj(...ReIm(...ZW(-y + pi / 4 + pi / 4, piecewise2(x, y, tau, 2), xdom))));
},
rangeX: [0 - 0.5, 3 * pi + 0.5],
rangeY: [0, pi / 4],
width: 250,
height: 30,
channels: 3,
items: 3,
});
view.surface({
id: 'insidesurface',
color: '#384452',
proximity: 10000000000,
shaded: false,
visible: true,
zBias: 10
});
view.line({
id: 'insidemesh',
color: '#00B3F3',
zBias: 10,
opacity: 0.2,
visible: true,
})
// Knot Curve
function knot(x) {
return (m2 * x) % (2 * pi);
}
view.interval({
range: [0, 2 * p * pi + 0.5],
expr: function(emit, x, i, t) {
emit(...SterProj(...ReIm(...ZW(π / 4, x % (2 * pi), knot(x)))));
},
width: 100,
channels: 4,
live: false
});
view.line({
id: 'knotcurve',
color: '#2ECC40',
proximity: 10,
blending: 'no',
zBias: 20
})
mathbox.set('focus', 2);
</script>
</div>
<!-- *****************************************************************************************************************
BLURB
***************************************************************************************************************** -->
<div class="info col-lg-4">
<h3>What is a Fibred Knot?</h3>
<p class="blurb">
A
<knot>knot</knot> \(K \in S^3 \) is <i>fibred</i> if there is a 1-parameter family \(F_\theta\) of <i>Seifert surfaces</i> for \(K\) called
<fibres>fibres</fibres> or
<fibres>fibration surfaces</fibres> where \(\theta \in [0,2\pi]\) runs through the points of a unit circle \(S^1\), such that if \(\theta\) is not equal to \(\theta^*\) then the intersection of their Fibres is exactly the knot, \(F_\theta \cap F_{\theta^*} = K\) .
</p>
<p class="blurb">
Therefore, we have a fibration map: $$f:S^3-K \rightarrow S^1$$ such that for each point on the cirle, \(\theta \in S^1\), the inverse fibration map, \(f^{-1}(\theta)\) , is a surface (3-manifold) with boundary \(K\).
</p>
<h3>Visualizing the Fibrations</h3>
<p class="blurb">
On this page we visualize the fibration of a
<knot>trefoil knot</knot>. Notice that the boundary of our
<fibres>fibration surface</fibres> is always exactly the knot. This visualization shows members of the family of fibres\(F_\theta\) by animating through values of \(\theta \in S^1\).
</p>
<p class="blurb">
In the subsequent pages we will understand and visualize how this fibration was constructed.
</p>
<nav aria-label="...">
<ul class="pager">
<li><a href="torusknot.html">Next</a></li>
<!-- <li><a href="#">Next</a></li>-->
</ul>
</nav>
</div>
</div>
<! --/row -->
</div>
<! --/container -->
<!-- *****************************************************************************************************************
FOOTER
***************************************************************************************************************** -->
<div id="footerwrap">
<div class="container">
<div class="row">
<div class="col-lg-4">
<h4>About Project</h4>
<div class="hline-w"></div>
<p>This is my final deliverable for the summer research component of a Master's in Mathematics at the University of Toronto supervised by Dr. Dror Bar-Natan. </p>
</div>
<div class="col-lg-4">
<h4>Project Repository</h4>
<div class="hline-w"></div>
<p style="text-align:center;">
<a href="http://www.github.com/jessebett/TorusKnotFibration"><i class="fa fa-github" style="font-size: 80px;"></i></a>
</p>
</div>
<div class="col-lg-4">
<h4>About Me</h4>
<div class="hline-w"></div>
<p>
My name is Jesse Bettencourt and I've recently acquired an M.Sc. in Mathematics from U of T. I'm interested in vizualizations from the intersection of math and computer science.
</p>
</div>
</div>
<! --/row -->
</div>
<! --/container -->
</div>
<! --/footerwrap -->
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