L11n47

L11n46

L11n48

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Link Presentations

[edit Notes on L11n47's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,14,6,15 X3849 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X17,2,18,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, -11, -2, 3, 8, -10, 9}

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^5-2 u v^4+2 u v^3-2 u v^2+u v+v^4-2 v^3+2 v^2-2 v+2}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{2}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{5}{q^{21/2}}-\frac{6}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{3}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-2 a^{13} z^{-1} +a^{11} z^5+5 a^{11} z^3+8 a^{11} z+4 a^{11} z^{-1} -a^9 z^7-5 a^9 z^5-7 a^9 z^3-3 a^9 z-a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-6 a^7 z-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^2 a^{16}+2 a^{16}-z^5 a^{15}-z^3 a^{15}-z a^{15}-2 z^6 a^{14}+2 z^4 a^{14}-3 z^2 a^{14}+a^{14}-2 z^7 a^{13}+z^5 a^{13}+6 z^3 a^{13}-8 z a^{13}+2 a^{13} z^{-1} -2 z^8 a^{12}+6 z^6 a^{12}-12 z^4 a^{12}+17 z^2 a^{12}-6 a^{12}-z^9 a^{11}+3 z^7 a^{11}-9 z^5 a^{11}+22 z^3 a^{11}-16 z a^{11}+4 a^{11} z^{-1} -3 z^8 a^{10}+12 z^6 a^{10}-17 z^4 a^{10}+15 z^2 a^{10}-5 a^{10}-z^9 a^9+4 z^7 a^9-5 z^5 a^9+4 z^3 a^9-3 z a^9+a^9 z^{-1} -z^8 a^8+4 z^6 a^8-3 z^4 a^8-2 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+6 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             2 1     -1 -14           4 2       2 -16         2 3         1 -18       4 3           1 -20     1 2             1 -22   2 4               -2 -24   1                 1 -26 2                   -2

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}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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