L11a119

L11a118

L11a120

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Link Presentations

[edit Notes on L11a119's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,11,21,12 X10,21,11,22 X16,10,17,9 X12,16,13,15 X8,18,9,17 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 7, -6, 5, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -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{(v-2) (2 v-1) (u v-2 u-2 v+1)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+7 q^{5/2}-12 q^{3/2}+16 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -3 z a^5-a^5 z^{-1} +3 z^3 a^3-a^3 z^{-1} -z^5 a+2 z^3 a+4 z a+2 a z^{-1} -z^5 a^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} +z^3 a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -4 a^4 z^8-9 a^2 z^8-6 z^8 a^{-2} -11 z^8-3 a^5 z^7-a^3 z^7+8 a z^7+2 z^7 a^{-1} -4 z^7 a^{-3} -2 a^6 z^6+5 a^4 z^6+25 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +35 z^6-a^7 z^5+2 a^5 z^5+8 a^3 z^5+10 a z^5+16 z^5 a^{-1} +11 z^5 a^{-3} +3 a^6 z^4-6 a^4 z^4-30 a^2 z^4-12 z^4 a^{-2} +2 z^4 a^{-4} -35 z^4+3 a^7 z^3+3 a^5 z^3-12 a^3 z^3-23 a z^3-18 z^3 a^{-1} -7 z^3 a^{-3} +5 a^4 z^2+16 a^2 z^2+4 z^2 a^{-2} +15 z^2-3 a^7 z-3 a^5 z+7 a^3 z+13 a z+6 z a^{-1} -a^6-2 a^4-3 a^2-1+a^7 z^{-1} +a^5 z^{-1} -a^3 z^{-1} -2 a z^{-1} - a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   4 1   -3 4                 8 3     5 2               8 4       -4 0             9 8         1 -2           9 9           0 -4         6 8             -2 -6       4 9               5 -8     2 6                 -4 -10   1 5                   4 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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