Three Dimensional Invariants: Difference between revisions

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<!--$$Select[AllKnots[],
<!--$$Select[AllKnots[],
(SymmetryType[#] == FullyAmphicheiral) &, 1]$$-->
(SymmetryType[#] == FullyAmphicheiral)
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 4 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == FullyAmphicheiral) &, 1]</nowiki> |
out= <nowiki>{Knot[4, 1]}</nowiki>}}
<!--END-->

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:

<!--$$Select[AllKnots[],
(SymmetryType[#] == Reversible) &, 1]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 5 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == Reversible) &, 1]</nowiki> |
out= <nowiki>{Knot[3, 1]}</nowiki>}}
<!--END-->

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

<!--$$Select[AllKnots[],
(SymmetryType[#] == NegativeAmphicheiral) &, 1]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 6 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == NegativeAmphicheiral) &, 1]</nowiki> |
out= <nowiki>{Knot[8, 17]}</nowiki>}}
<!--END-->

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

<!--$$Select[AllKnots[],
(SymmetryType[#] == Chiral) &, 1]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 7 |
in = <nowiki>Select[AllKnots[],
(SymmetryType[#] == Chiral) &, 1]</nowiki> |
out= <nowiki>{Knot[9, 32]}</nowiki>}}
<!--END-->

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

<!--$$Plus @@ (SymmetryType /@ Rest[AllKnots[]])$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 8 |
in = <nowiki>Plus @@ (SymmetryType /@ Rest[AllKnots[]])</nowiki> |
out= <nowiki>216 Chiral + 13 FullyAmphicheiral + 7 NegativeAmphicheiral +
565 Reversible</nowiki>}}
<!--END-->

{{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>.

<!--$$?UnknottingNumber$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

<!--$UH = Plus @@ u /@ Cases[UnknottingNumber /@ AllKnots[], _Integer];$--><!--END-->

Of the <!--$UH /. _u -> 1$--><!--Robot Land, no human edits to "END"-->512<!--END--> knots whose unknotting number is known to <code>KnotTheory`</code>, <!--$Coefficient[UH, u[1]]$--><!--Robot Land, no human edits to "END"-->197<!--END--> have unknotting number 1, <!--$Coefficient[UH, u[2]]$--><!--Robot Land, no human edits to "END"-->247<!--END--> have unknotting number 2, <!--$Coefficient[UH, u[3]]$--><!--Robot Land, no human edits to "END"-->54<!--END--> have unknotting number 3, <!--$Coefficient[UH, u[4]]$--><!--Robot Land, no human edits to "END"-->12<!--END--> have unknotting number 4 and <!--$Coefficient[UH, u[5]]$--><!--Robot Land, no human edits to "END"-->1<!--END--> has unknotting number 5:

<!--$$Plus @@ u /@ Cases[UnknottingNumber /@ AllKnots[], _Integer]$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

There are <!--$Length[Select[AllKnots[], Crossings[#] <= 9 && Head[UnknottingNumber[#]] === List &]
]$--><!--Robot Land, no human edits to "END"-->4<!--END--> knots with up to 9 crossings whose unknotting number is unknown:

<!--$$Select[AllKnots[],
Crossings[#] <= 9 && Head[UnknottingNumber[#]] === List &
]$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

{{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 [http://www.win.tue.nl/~vanwijk/seifertview/ SeifertView programme] 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.


<!--$$?ThreeGenus$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

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:

<!--$$Max[ThreeGenus /@ AllKnots[]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 15 |
in = <nowiki>Max[ThreeGenus /@ AllKnots[]]</nowiki> |
out= <nowiki>5</nowiki>}}
<!--END-->

<!--$$Select[AllKnots[], ThreeGenus[#] == 5 &]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 16 |
in = <nowiki>Select[AllKnots[], ThreeGenus[#] == 5 &]</nowiki> |
out= <nowiki>{Knot[11, Alternating, 367]}</nowiki>}}
<!--END-->

{{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]]:

<!--$$ThreeGenus[BR[4, {1, 1, 2, -3, 2, 1, -3, -2, -2, -3, -3}]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 17 |
in = <nowiki>ThreeGenus[BR[4, {1, 1, 2, -3, 2, 1, -3, -2, -2, -3, -3}]]</nowiki> |
out= <nowiki>3</nowiki>}}
<!--END-->

<!--$$ThreeGenus[Knot[11, NonAlternating, 42]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 18 |
in = <nowiki>ThreeGenus[Knot[11, NonAlternating, 42]]</nowiki> |
out= <nowiki>2</nowiki>}}
<!--END-->

{{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>.

<!--$$?BridgeIndex$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

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:

<!--$$Select[AllKnots[], Crossings[#] == 9 && BridgeIndex[#] == 2 &]$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

====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.

<!--$$?SuperBridgeIndex$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

====Nakanishi Index====

<!--$$?NakanishiIndex$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

====Synthesis====

<!--$$Profile[K_] := Profile[
SymmetryType[K], UnknottingNumber[K], ThreeGenus[K],
BridgeIndex[K], SuperBridgeIndex[K], NakanishiIndex[K]
]$$-->
<!--Robot Land, no human edits to "END"-->
{{In|
n = 26 |
in = <nowiki>Profile[K_] := Profile[
SymmetryType[K], UnknottingNumber[K], ThreeGenus[K],
BridgeIndex[K], SuperBridgeIndex[K], NakanishiIndex[K]
]</nowiki>}}
<!--END-->

<!--$$Profile[Knot[9,24]]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 27 |
in = <nowiki>Profile[Knot[9,24]]</nowiki> |
out= <nowiki>Profile[Reversible, 1, 3, 3, {4, 6}, 1]</nowiki>}}
<!--END-->

<!--$$Ks = Select[AllKnots[], (Crossings[#] == 9 && Profile[#]==Profile[Knot[9,24]])&]$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

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

<!--$$Alexander[#][t]& /@ Ks$$-->
<!--Robot Land, no human edits to "END"-->
{{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>}}
<!--END-->

Revision as of 09:13, 15 June 2007


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

Symmetry Type

In[2]:= ?SymmetryType
SymmetryType[K] returns the symmetry type of the knot K, if known to KnotTheory`. The possible types are: Reversible, FullyAmphicheiral, NegativeAmphicheiral and Chiral.
In[3]:= SymmetryType::about
The symmetry type data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

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

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


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