L11a7

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Link Presentations

[edit Notes on L11a7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,5,13 X14,22,15,21 X18,10,19,9 X20,12,21,11 X10,20,11,19 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 8, -10, 9, -4, 6, -7, 11, -2, 3, -8, 10, -9, 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{(u-1) (v-1) \left(2 v^4-3 v^3+3 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 16 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-5 q^{15/2}+10 q^{13/2}-14 q^{11/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-7} -3 z^3 a^{-7} -z a^{-7} + a^{-7} z^{-1} +z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +z^3 a^{-3} -z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} -2 z^3 a^{-11} +3 z^6 a^{-10} -7 z^4 a^{-10} +4 z^2 a^{-10} +4 z^7 a^{-9} -7 z^5 a^{-9} +3 z^3 a^{-9} +4 z^8 a^{-8} -4 z^6 a^{-8} -3 z^4 a^{-8} +7 z^2 a^{-8} -2 a^{-8} +4 z^9 a^{-7} -8 z^7 a^{-7} +13 z^5 a^{-7} -9 z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} -2 z^6 a^{-6} +z^4 a^{-6} +6 z^2 a^{-6} -5 a^{-6} +9 z^9 a^{-5} -27 z^7 a^{-5} +38 z^5 a^{-5} -24 z^3 a^{-5} +3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +2 z^8 a^{-4} -12 z^6 a^{-4} +8 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -11 z^7 a^{-3} +6 z^5 a^{-3} -4 z^3 a^{-3} +2 z a^{-3} +6 z^8 a^{-2} -16 z^6 a^{-2} +9 z^4 a^{-2} -z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +6 z^3 a^{-1} +z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4 }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     2   -2 16                   3 1   2 14                 7 2     -5 12               7 3       4 10             9 7         -2 8           8 7           1 6         6 9             3 4       6 8               -2 2     3 8                 5 0   1 4                   -3 -2   3                     3 -4 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/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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