In mathematics, the knot complement of a tame knot K is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N, The knot complement XK is a compact 3-manifold; the boundary of XK and the boundary of the neighborhood N are homeomorphic to a two-torus. Sometimes the ambient manifold M is understood to be 3-sphere. Context is needed to determine the usage. There are analogous definitions of link complement. Many knot invariants, such as the knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the Gordon–Luecke theorem states that a knot is determined by its complement. That is, if K and K′ are two knots with homeomorphic complements then there is a homeomorphism of the three-sphere taking one knot to the other. (Wikipedia).
This step by step guide demonstrates tying 15 types: 00:36 Overhand 01:22 Square 02:36 Figure Eight 03:40 Bowline, 05:29 Running 06:19 Half, 07:45 Timber, 09:42 Rolling, 10:43 Clove Hitches 11:30 Cat's Paw 12:58 Single, 14:40 Double Sheet or Becket Bends 15:30 Fisherman's, 17:09 Doubl
From playlist How To Tutorials
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/9Wje This is joint work with François Guéritaud and Saul Schleimer.
From playlist 3D printing
Algebraic topology: Fundamental group of a knot
This lecture is part of an online course on algebraic topology. We calculate the fundamental group of (the complement of) a knot, and give a couple of examples. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52yxQGxQoxwOtjIEtxE2BWx
From playlist Algebraic topology
This knot is very useful for adjusting tie downs quickly and easily. For example, a tarp could be held down by a series of these knots and be made very tight so the wind cannot make it rise, and easily be removed simply by sliding the knots later. A taut line knot is also used to keep larg
From playlist Practical Projects & Skills
Link: https://www.geogebra.org/m/a72HSgzU
From playlist Geometry: Challenge Problems
Untangling the beautiful math of KNOTS
Visit ► https://brilliant.org/TreforBazett/ to help you learn STEM topics for free, and the first 200 people will get 20% off an annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor Suppose yo
From playlist Cool Math Series
Link: https://www.geogebra.org/m/bd69d6u4
From playlist Geometry: Challenge Problems
Unofoil with cogs: http://shpws.me/wk7u Trefoil with cogs: http://shpws.me/wk7H Cinquefoil with cogs: http://shpws.me/wk7t
From playlist 3D printing
Rima Chatterjee: Structure Theorems of Legendrian Knots in Contact Manifolds
Rima Chatterjee, University of Cologne Title: Structure Theorems of Legendrian Knots in Contact Manifolds Structure theorems of a Legendrian knot have been an interesting study in contact geometry. One can ask when topological operations on a knot gives us important information about the g
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Rebekah Palmer: Totally Geodesic Surfaces in Knot Complements with Small Crossing Number
Rebekah Palmer, Temple University Title: Totally Geodesic Surfaces in Knot Complements with Small Crossing Number Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3-manifolds. Recently, Bader-Fisher-Miller-Stover showed that con
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Cannon-Thurston maps: naturally occurring space-filling curves
Saul Schleimer and I attempt to explain what a Cannon-Thurston map is. Thanks to my brother Will Segerman for making the carvings, and to Daniel Piker for making the figure-eight knot animations. I made the animation of the (super crinkly) surface using our app (with Dave Bachman) for coh
From playlist GPU shaders
Link: https://www.geogebra.org/m/JEk3MHvc
From playlist Geometry: Challenge Problems
Jessica Purcell - Lecture 1 - Hyperbolic knots and alternating knots
Jessica Purcell, Monash University Title: Hyperbolic knots and alternating knots Hyperbolic geometry has been used since around the mid-1970s to study knot theory, but it can be difficult to relate geometry of knots to a diagram of a knot. However, many results from the 1980s and beyond s
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Hyperbolic Knot Theory (Lecture - 2) by Abhijit Champanerkar
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)
Cabling of knots in overtwisted contact manifolds - Rima Chatterjee
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Title: Cabling of knots in overtwisted contact manifolds Speaker: Rima Chatterjee Affiliation: Cologne Date: October 8, 2021 Abstract: Knots associated to overtwisted manifolds are less explored. There are two types of kno
From playlist Mathematics
Graham ELLIS - Computational group theory, cohomology of groups and topological methods 3
The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP. Alexander Hulpke's lectures will being with some general computation
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Link: https://www.geogebra.org/m/cjS3b6Zr
From playlist Geometry: Challenge Problems
Colin Adams: Hyperbolic Volumes of Virtual Knots and their Compositions
Colin Adams, Williams College Title: Hyperbolic Volumes of Virtual Knots and their Compositions Hyperbolic Volumes of Virtual Knots and their Compositions\\ \noindent\textbf{Abstract:} Many knots are known to be hyperbolic and therefore have a hyperbolic volume. But composite knots are n
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022