L11a71

L11a70

L11a72

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Link Presentations

[edit Notes on L11a71's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X22,20,5,19 X20,14,21,13 X14,22,15,21 X10,16,11,15 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, 7, -8, 9, -3, 5, -4, 6, -7, 8, -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{u v^5-4 u v^4+7 u v^3-8 u v^2+6 u v-3 u-3 v^5+6 v^4-8 v^3+7 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 17 q^{9/2}-19 q^{7/2}+18 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-4 q^{15/2}+9 q^{13/2}-13 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -4 z^3 a^{-5} + a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-5 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -9 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +12 z^7 a^{-7} -16 z^5 a^{-7} +8 z^3 a^{-7} -2 z a^{-7} +10 z^8 a^{-6} -9 z^6 a^{-6} -5 z^4 a^{-6} +3 z^2 a^{-6} +5 z^9 a^{-5} +7 z^7 a^{-5} -32 z^5 a^{-5} +20 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^{10} a^{-4} +16 z^8 a^{-4} -46 z^6 a^{-4} +38 z^4 a^{-4} -15 z^2 a^{-4} +3 a^{-4} +8 z^9 a^{-3} -14 z^7 a^{-3} -3 z^5 a^{-3} +5 z^3 a^{-3} +3 z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +8 z^8 a^{-2} -34 z^6 a^{-2} +36 z^4 a^{-2} -10 z^2 a^{-2} +3 z^9 a^{-1} +a z^7-8 z^7 a^{-1} -5 a z^5+4 z^5 a^{-1} +9 a z^3+3 z^3 a^{-1} -7 a z-4 z a^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} +2 z^8-6 z^6+3 z^4+4 z^2-3 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 7 3     4 10               10 6       -4 8             9 7         2 6           9 10           1 4         6 9             -3 2       5 10               5 0     2 5                 -3 -2     5                   5 -4 1 2                     -1 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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