Editing Knot theory (section)

==Knot equivalence<!--[[Knot equivalence]] redirects directly here.-->==
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A knot is created by beginning with a one-[[dimension]]al line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop {{Harv|Adams|2004}} {{Harv|Sossinsky|2002}}. Simply, we can say a knot <math>K</math> is a "simple closed curve" (see [[Curve]]) — that is: a "nearly" [[injective]] and [[continuous function]] <math>K\colon[0,1]\to \mathbb{R}^3</math>, with the only "non-injectivity" being <math>K(0)=K(1)</math>. Topologists consider knots and other entanglements such as [[link (knot theory)|links]] and [[Braid theory|braid]]s to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.

The idea of '''knot equivalence''' is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots <math>K_1, K_2</math> are equivalent if there is an [[orientation-preserving]] [[homeomorphism]] <math>h\colon\R^3\to\R^3</math> with <math>h(K_1)=K_2</math>.

What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms <math>\{ h_t: \mathbb R^3 \rightarrow \mathbb R^3\ \mathrm{for}\ 0 \leq t \leq 1\}</math> of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail:  Two knots <math>K_1</math> and <math>K_2</math> are '''equivalent''' if there exists a continuous mapping <math>H: \mathbb R^3 \times [0,1] \rightarrow \mathbb R^3</math> such that a) for each <math>t \in [0,1]</math> the mapping taking <math>x \in \mathbb R^3</math> to <math>H(x,t) \in \mathbb R^3</math> is a homeomorphism of <math>\mathbb R^3</math> onto itself; b) <math>H(x, 0) = x</math> for all <math>x \in \mathbb R^3</math>; and c) <math>H(K_1,1) = K_2</math>. Such a function <math>H</math> is known as an [[ambient isotopy]].)

These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of <math>\mathbb R^3</math> to itself is the final stage of an ambient isotopy starting from the identity. Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the <math>t=1</math> (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.

The basic problem of knot theory, the '''recognition problem''', is determining the equivalence of two knots.  [[Algorithm]]s exist to solve this problem, with the first given by [[Wolfgang Haken]] in the late 1960s {{Harv|Hass|1998}}. Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is {{Harv|Hass|1998}}. The special case of recognizing the [[unknot]], called the [[unknotting problem]], is of particular interest {{Harv|Hoste|2005}}. In February 2021 [[Marc Lackenby]] announced a new unknot recognition algorithm that runs in [[Time complexity|quasi-polynomial time]].<ref>{{citation|url=https://www.maths.ox.ac.uk/node/38304|title=Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time|date=2021-02-03|publisher=Mathematical Institute, [[University of Oxford]]|accessdate=2021-02-03}}</ref>