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...y and [[3-manifold]] theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame. Knots that are not tame are called '''wild''' and can have [[pathological (mathematics)|pathological]] behavior. ...

{{short description|Family of mathematical knots}} ...s are an infinite family of knots, and are considered the simplest type of knots after the [[torus knot]]s. ...

{{short description|Connected sum of two trefoil knots with same chirality}} ...not, the granny knot and the square knot are the simplest of all composite knots. ...

...[[Wilhelm Wirtinger]] observed that the [[knot complement|complements of knots]] in [[Three-dimensional space|3-space]] have [[fundamental group]]s with p ==Preliminaries and definition== ...

...sted as the '''6₁ knot''' in the [[Alexander–Briggs notation]], and it can also be described as a [[twist knot]] with four half twists, or as t and its [[Jones polynomial]] is ...

...projection onto the plane perpendicular to ''v'' gives a [[knot diagram]], and we can compute the crossing number, denoted ''n''(''v''). The average cross ...where projection doesn't give a knot diagram is a set of [[measure zero]] and ''n''(''v'') is [[locally constant]] when defined. ...

...le=The knot book: an elementary introduction to the mathematical theory of knots |publisher=American Mathematical Society |location=Providence, Rhode Island | journal = Journal of Knot Theory and its Ramifications ...

...ually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures. ...hexagram can appear in non-interlaced versions, in which case they are not knots at all.) ...

{{short description|A mathematical operator used in theoretical physics and topology}} ..."Kauffman1991">Kauffman, L.H. (1991). "Knots and physics". Series on Knots and Everything. 1. World Scientific. ISBN 978-981-02-0332-1.</ref> ...

{{Short description|Conjecture in knot theory relating quantum invariants and hyperbolic geometry}} * all knots and links with volume zero ...

{{short description|Gives necessary and sufficient conditions for two braids to have equivalent closures}} In [[mathematics]] the '''Markov theorem''' gives necessary and sufficient conditions for two [[braid (mathematics)|braids]] to have closur ...

...ine the '''Petaluma model''', a family of [[probability distribution]]s on knots with a given number of petals, defined by choosing a [[random permutation]] ...ssing are not allowed. The "petals" formed by arcs of the curve that leave and then return to this crossing point are all non-nested, bounding closed disk ...

...six, the others being the [[Stevedore knot (mathematics)|stevedore knot]] and the [[63 knot|6₃ knot]]. This knot is sometimes referred to as and its [[Jones polynomial]] is ...

...]. It is [[alternating knot|alternating]], [[hyperbolic knot|hyperbolic]], and [[amphichiral knot|fully amphichiral]]. and the [[Kauffman polynomial]] is ...

...e 52 knot.}}</ref> in the [[Alexander-Briggs notation]], and is one of two knots with [[crossing number (knot theory)|crossing number]] five, the other bein The three-twist knot is a [[prime knot]], and it is [[invertible knot|invertible]] but not [[amphichiral knot|amphichiral ...

...ref> The concept is useful because a sailboat cannot sail directly upwind, and thus often can not, or should not, sail directly to a mark to reach it as q ...ctly what the optimum angle against the wind is. At the optimum boat speed and angle to the wind, VMG is maximized, steering closer to the direction of th ...

| caveat=Not secure as a bend unless secured by additional knots(ex: overhand). [[Knot#Capsizing|Spills]] easily if one of the free ends is ...hich is often appended with the rhyming suffix "... makes a knot both tidy and tight". Two consecutive overhands tied as described above of the same [[h ...

...formal sum]] of [[Jacobi diagram]]s (which carry a Lie algebra structure), and a representation of a [[ribbon Hopf algebra]] such as a [[quantum group]]. ...riant of [[Kirby calculus]] on which only certain specific types of framed links are allowed. Claspers may also be interpreted algebraically, as a [[diagram ...

...n low-dimensional [[geometric topology]] (specifically of 4-[[manifold]]s) and the mathematics of [[gauge theory]]. ==Education and career== ...

| death_date ={{death date and age |2014|12|16 |1955|04|07 |}} ...[low-dimensional topology]], [[knot theory|the theory of knots and links]] and associated algebra. ...