L11n453

L11n452

L11n454

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Link Presentations

[edit Notes on L11n453's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {-4, -5, 11, 9, -6, 8, -7, 6}

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 v w^2 x^2-u v w^2 x-u v w x^2-u w^2 x^2+u w^2 x+u w x^2-u w x-v w x+v w+v x-v-w-x+2}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{6}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}+a^{13} z^{-3} +z^5 a^{11}+4 z^3 a^{11}+z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} -z^7 a^9-4 z^5 a^9+11 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -z^7 a^7-6 z^5 a^7-12 z^3 a^7-11 z a^7-5 a^7 z^{-1} -a^7 z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{16}-3 z^3 a^{15}+3 z a^{15}-z^6 a^{14}+z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+12 z^5 a^{13}-18 z^3 a^{13}+12 z a^{13}-5 a^{13} z^{-1} +a^{13} z^{-3} -3 z^8 a^{12}+12 z^6 a^{12}-12 z^4 a^{12}-3 z^2 a^{12}-3 a^{12} z^{-2} +10 a^{12}-z^9 a^{11}+14 z^5 a^{11}-25 z^3 a^{11}+21 z a^{11}-12 a^{11} z^{-1} +3 a^{11} z^{-3} -4 z^8 a^{10}+16 z^6 a^{10}-10 z^4 a^{10}-15 z^2 a^{10}-6 a^{10} z^{-2} +19 a^{10}-z^9 a^9+2 z^7 a^9+8 z^5 a^9-22 z^3 a^9+23 z a^9-12 a^9 z^{-1} +3 a^9 z^{-3} -z^8 a^8+3 z^6 a^8+3 z^4 a^8-12 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+11 z a^7-5 a^7 z^{-1} +a^7 z^{-3} }[/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               3     3 -12           1 1 1     1 -14           6 3       3 -16       2 2 2         2 -18       6 4           2 -20   1 2 4             3 -22   2 2               0 -24   2                 2 -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{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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