L11n151

L11n150

L11n152

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Link Presentations

[edit Notes on L11n151's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 u^2 v^4-2 u^2 v^3+u^2 v^2+u v^3-u v^2+u v+v^2-2 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{3}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+5 z^3 a^{11}+6 z a^{11}+2 a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-6 z^3 a^9-z a^9-z^7 a^7-6 z^5 a^7-11 z^3 a^7-7 z a^7-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{16}-2 z^3 a^{15}+z a^{15}-3 z^4 a^{14}+4 z^2 a^{14}-2 a^{14}-z^7 a^{13}+2 z^5 a^{13}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+10 z^6 a^{12}-20 z^4 a^{12}+19 z^2 a^{12}-5 a^{12}-z^9 a^{11}+4 z^7 a^{11}-7 z^5 a^{11}+12 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} -3 z^8 a^{10}+14 z^6 a^{10}-19 z^4 a^{10}+11 z^2 a^{10}-3 a^{10}-z^9 a^9+4 z^7 a^9-3 z^5 a^9-z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+7 z a^7-a^7 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                   1 1 -8                 1 1 0 -10               2     2 -12             1 1     0 -14           3 2       1 -16         1 1         0 -18       3 3           0 -20     1 2             1 -22   1 2               -1 -24   1                 1 -26 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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