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The simplest example of a non-invertible knot is the knot 8 17 (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2 p + 1), (2 q + 1), (2 r + 1), where p , q , and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.
A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10.
A framed knot is the extension of a tame knot to an embedding of the solid torus D 2 × S 1 in S 3. The framing of the knot is the linking number of the image of the ribbon I × S 1 with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. [8] This definition generalizes to an analogous ...
In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway. [1]It is related by mutation to the Kinoshita–Terasaka knot, [3] with which it shares the same Jones polynomial.
80 is: the sum of Euler's totient function φ( x ) over the first sixteen integers. a semiperfect number , since adding up some subsets of its divisors (e.g., 1 , 4 , 5 , 10 , 20 and 40 ) gives 80.
The discovery of all possible ways to tie a tie depends on a mathematical formulation of the act of tying a tie. In their papers (which are technical) and book (which is for a lay audience, apart from an appendix), the authors show that necktie knots are equivalent to persistent random walks on a triangular lattice, with some constraints on how the walks begin and end.
The figure-eight loop is used like an overhand loop knot. This type of knot can be used in prusik climbing when used in conjunction with a climbing harness, a climbing rope, and locking carabiner designed for climbing, to ascend or descend with minimal equipment and effort.
The directional figure eight (a.k.a. inline figure-eight loop) is a loop knot.It is a knot that can be made on the bight.The loop must only be loaded in the correct direction or the knot may fail.