User:Ranicki

2006-04-16T11:01:42Z

Ranicki: Information about Andrew Ranicki

I am Andrew Ranicki from the University of Edinburgh.
My home page
http://www.maths.ed.ac.uk/~aar
has a directory
http://www.maths.ed.ac.uk/~aar/knots
devoted to knot theory, particularly the early history.

What's New?

2006-04-16T10:59:22Z

Ranicki:

This is a reverse chronological list of significant changes/additions to the Knot Atlas.

* A link to the SeifertView visualization programme of Jack van Wijk for Seifert surfaces. --[[User:ranicki|ranicki]] 6:57, 16 Apr 2006 (EDT)
* The Knot Atlas contains values for the Rasmussen s-Invariant up to 11 crossings, taken from KnotInfo. --[[User:Scott|Scott]] 19:32, 13 Apr 2006 (EDT)
* Thanks to Jake Rasmussen, [[The Mathematica Package KnotTheory`|KnotTheory`]] can now compute the [[Three_Dimensional_Invariants#3-Genus|ThreeGenus]] for many knots.
--[[User:Drorbn|Drorbn]] 06:59, 4 Feb 2006 (EST)
* [[Tour of the Knot Atlas]] added.
--[[User:Drorbn|Drorbn]] 21:24, 22 Sep 2005 (EDT)
* All Rolfsen knots now show a link to a 'more quantum invariants page'.
--[[User:Scott|Scott]] 19:29, 15 Sep 2005 (EDT)
* The multivariable Alexander polynomial appears on all link pages.
--[[User:Drorbn|Drorbn]] 09:43, 12 Sep 2005 (EDT)
* Some data from Livingston's KnotInfo has been imported, but is not yet displayed on any knot pages. The wiki now contains hyperbolic volumes, tau invariants (except for non-alternating 11 crossing knots), and conway notations. See for example [[Data:K11a5/HyperbolicVolume]], [[Data:K11a308/TauInvariant]] and [[Data:6_2/ConwayNotation]]. What else do we want from KnotInfo?
--[[User:Scott|Scott]] 14:19, 7 Sep 2005 (EDT)
* The multivariable Alexander polynomial is in, at [[The Multivariable Alexander Polynomial]].
--[[User:Drorbn|Drorbn]] 16:00, 5 Sep 2005 (EDT)
* Vogel's algorithm is in, at [[Braid Representatives]]; braid representative are shown on all knot pages.
--[[User:Drorbn|Drorbn]] 12:04, 3 Sep 2005 (EDT)

Three Dimensional Invariants

2006-04-16T10:55:53Z

Ranicki: /* 3-Genus */

{{Manual TOC Sidebar}}

{{Startup Note}}

====Symmetry Type====



{{HelpAndAbout|
n = 2 |
n1 = 3 |
in = <nowiki>SymmetryType</nowiki> |
out= <nowiki>SymmetryType[K] returns the symmetry type of the knot K, if known to KnotTheory`. The possible types are: Reversible, FullyAmphicheiral, NegativeAmphicheiral and Chiral.</nowiki> |
about= <nowiki>The symmetry type data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}


The inverse of a knot <math>K</math> is the knot obtained from it by reversing its parametrization. The mirror of A knot <math>K</math> is obtained from <math>K</math> by reversing the orientation of the ambient space, or, alternatively, by flipping all the crossings of <math>K</math>.

A knot is called "fully amphicheiral" if it is equal to its inverse and also to its mirror. The first knot with this property is



{{InOut|
n = 4 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == FullyAmphicheiral) &, 1]</nowiki> |
out= <nowiki>{Knot[4, 1]}</nowiki>}}


A knot is called "reversible" if it is equal to its inverse yet it different from its mirror (and hence also from the inverse of its mirror). Many knots have this property; indeed, the first one is:



{{InOut|
n = 5 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == Reversible) &, 1]</nowiki> |
out= <nowiki>{Knot[3, 1]}</nowiki>}}


A knot is called "positive amphicheiral" if it is different from its inverse but equal to its mirror. There are no such knots with up to 11 crossings.

A knot is called "negative amphicheiral" if it is different from its inverse and its mirror, yet it is equal to the inverse of its mirror. The first knot with this property is



{{InOut|
n = 6 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == NegativeAmphicheiral) &, 1]</nowiki> |
out= <nowiki>{Knot[8, 17]}</nowiki>}}


Finally, if a knot is different from its inverse, its mirror and from the inverse of its mirror, it is "chiral". The first such knot is



{{InOut|
n = 7 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == Chiral) &, 1]</nowiki> |
out= <nowiki>{Knot[9, 32]}</nowiki>}}


It is a amusing to take "symmetry type" statistics on all the prime knots with up to 11 crossings:



{{InOut|
n = 8 |
in = <nowiki>Plus @@ (SymmetryType /@ Rest[AllKnots[]])</nowiki> |
out= <nowiki>216 Chiral + 13 FullyAmphicheiral + 7 NegativeAmphicheiral +

565 Reversible</nowiki>}}


{{Knot Image Quadruple|4_1|gif|3_1|gif|8_17|gif|9_32|gif}}

====Unknotting Number====

The ''unknotting number'' of a knot <math>K</math> is the minimal number of crossing changes needed in order to unknot <math>K</math>.



{{HelpAndAbout|
n = 9 |
n1 = 10 |
in = <nowiki>UnknottingNumber</nowiki> |
out= <nowiki>UnknottingNumber[K] returns the unknotting number of the knot K, if known to KnotTheory`. If only a range of possible values is known, a list of the form {min, max} is returned.</nowiki> |
about= <nowiki>The unknotting numbers of torus knots are due to ???. All other unknotting numbers known to KnotTheory` are taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}




Of the 512 knots whose unknotting number is known to <code>KnotTheory`</code>, 197 have unknotting number 1, 247 have unknotting number 2, 54 have unknotting number 3, 12 have unknotting number 4 and 1 has unknotting number 5:



{{InOut|
n = 11 |
in = <nowiki>Plus @@ u /@ Cases[UnknottingNumber /@ AllKnots[], _Integer]</nowiki> |
out= <nowiki>u[0] + 197 u[1] + 247 u[2] + 54 u[3] + 12 u[4] + u[5]</nowiki>}}


There are 4 knots with up to 9 crossings whose unknotting number is unknown:



{{InOut|
n = 12 |
in = <nowiki>Select[AllKnots[],
Crossings[#] <= 9 && Head[UnknottingNumber[#]] === List &
]</nowiki> |
out= <nowiki>{Knot[9, 10], Knot[9, 13], Knot[9, 35], Knot[9, 38]}</nowiki>}}


{{Knot Image Quadruple|9_10|gif|9_13|gif|9_35|gif|9_38|gif}}

====3-Genus====

A Seifert surface for a knot <math>K \subset S^3</math> is a compact oriented surface <math>L \subset S^3</math>
with boundary <math>\partial L=K</math>. Seifert surfaces exist, but are not unique. The SeifertView programme
http://www.win.tue.nl/~vanwijk/seifertview/ is a visual implementation of the algorithm of Seifert (1934) for
the construction of a Seifert surface from a knot projection. The 3-genus of a knot is the minimal genus of a
Seifert surface for that knot.



{{HelpAndAbout|
n = 13 |
n1 = 14 |
in = <nowiki>ThreeGenus</nowiki> |
out= <nowiki>ThreeGenus[K] returns the 3-genus of the knot K or a list of the form {lower bound, upper bound}.</nowiki> |
about= <nowiki>The 3-genus program was written by Jake Rasmussen of Princeton University. The program tries to compute the highest nonvanishing group in the knot Floer homology, using Ozsvath and Szabo's version of the Kauffman state model.</nowiki>}}


The highest 3-genus of the knots known to <tt>KnotTheory`</tt> is <math>5</math>, and there is only one knot with up to 11 crossings whose 3-genus is 5:



{{InOut|
n = 15 |
in = <nowiki>Max[ThreeGenus /@ AllKnots[]]</nowiki> |
out= <nowiki>5</nowiki>}}




{{InOut|
n = 16 |
in = <nowiki>Select[AllKnots[], ThreeGenus[#] == 5 &]</nowiki> |
out= <nowiki>{Knot[11, Alternating, 367]}</nowiki>}}


{{Knot Image Pair|K11a367|gif|T(11,2)|jpg}}

([[K11a367]] is, of couse, also known as the torus knot [[T(11,2)]]).

The Conway knot [[K11n34]] is the closure of the braid <tt>BR[4, {1, 1, 2, -3, 2, 1, -3, -2, -2, -3, -3}]</tt>. Let us compute its 3-genus and compare it with the 3-genus of its mutant companion, the Kinoshita-Terasaka knot [[K11n42]]:



{{InOut|
n = 17 |
in = <nowiki>ThreeGenus[BR[4, {1, 1, 2, -3, 2, 1, -3, -2, -2, -3, -3}]]</nowiki> |
out= <nowiki>3</nowiki>}}




{{InOut|
n = 18 |
in = <nowiki>ThreeGenus[Knot[11, NonAlternating, 42]]</nowiki> |
out= <nowiki>2</nowiki>}}


{{Knot Image Pair|K11n34|gif|K11n32|gif}}

====Bridge Index====

The ''bridge index' of a knot <math>K</math> is the minimal number of local maxima (or local minima) in a generic smooth embedding of <math>K</math> in <math>{\mathbf R}^3</math>.



{{HelpAndAbout|
n = 19 |
n1 = 20 |
in = <nowiki>BridgeIndex</nowiki> |
out= <nowiki>BridgeIndex[K] returns the bridge index of the knot K, if known to KnotTheory`.</nowiki> |
about= <nowiki>The bridge index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}


An often studied class of knots is the class of 2-bridge knots, knots whose bridge index is 2. Of the 49 prime 9-crossings knots, 24 are 2-bridge:



{{InOut|
n = 21 |
in = <nowiki>Select[AllKnots[], Crossings[#] == 9 && BridgeIndex[#] == 2 &]</nowiki> |
out= <nowiki>{Knot[9, 1], Knot[9, 2], Knot[9, 3], Knot[9, 4], Knot[9, 5],

Knot[9, 6], Knot[9, 7], Knot[9, 8], Knot[9, 9], Knot[9, 10],

Knot[9, 11], Knot[9, 12], Knot[9, 13], Knot[9, 14], Knot[9, 15],

Knot[9, 17], Knot[9, 18], Knot[9, 19], Knot[9, 20], Knot[9, 21],

Knot[9, 23], Knot[9, 26], Knot[9, 27], Knot[9, 31]}</nowiki>}}


====Super Bridge Index====

The ''super bridge index'' of a knot <math>K</math> is the minimal number, in a generic smooth embedding of <math>K</math> in <math>{\mathbf R}^3</math>, of the maximal number of local maxima (or local minima) in a rigid rotation of that projection.



{{HelpAndAbout|
n = 22 |
n1 = 23 |
in = <nowiki>SuperBridgeIndex</nowiki> |
out= <nowiki>SuperBridgeIndex[K] returns the super bridge index of the knot K, if known to KnotTheory`. If only a range of possible values is known, a list of the form {min, max} is returned.</nowiki> |
about= <nowiki>The super bridge index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}


====Nakanishi Index====



{{HelpAndAbout|
n = 24 |
n1 = 25 |
in = <nowiki>NakanishiIndex</nowiki> |
out= <nowiki>NakanishiIndex[K] returns the Nakanishi index of the knot K, if known to KnotTheory`.</nowiki> |
about= <nowiki>The Nakanishi index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}


====Synthesis====



{{In|
n = 26 |
in = <nowiki>Profile[K_] := Profile[
SymmetryType[K], UnknottingNumber[K], ThreeGenus[K],
BridgeIndex[K], SuperBridgeIndex[K], NakanishiIndex[K]
]</nowiki>}}




{{InOut|
n = 27 |
in = <nowiki>Profile[Knot[9,24]]</nowiki> |
out= <nowiki>Profile[Reversible, 1, 3, 3, {4, 6}, 1]</nowiki>}}




{{InOut|
n = 28 |
in = <nowiki>Ks = Select[AllKnots[], (Crossings[#] == 9 && Profile[#]==Profile[Knot[9,24]])&]</nowiki> |
out= <nowiki>{Knot[9, 24], Knot[9, 28], Knot[9, 30], Knot[9, 34]}</nowiki>}}


{{Knot Image Quadruple|9_24|gif|9_28|gif|9_30|gif|9_34|gif}}



{{InOut|
n = 29 |
in = <nowiki>Alexander[#][t]& /@ Ks</nowiki> |
out= <nowiki> -3 5 10 2 3
{13 - t + -- - -- - 10 t + 5 t - t ,
2 t
t

-3 5 12 2 3
-15 + t - -- + -- + 12 t - 5 t + t ,
2 t
t

-3 5 12 2 3
17 - t + -- - -- - 12 t + 5 t - t ,
2 t
t

-3 6 16 2 3
23 - t + -- - -- - 16 t + 6 t - t }
2 t
t</nowiki>}}

Knot Atlas - User contributions [en]

User:Ranicki

What's New?

Three Dimensional Invariants