L11a371

L11a370

L11a372

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Link Presentations

[edit Notes on L11a371's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,7,21 X20,8,21,7 X18,10,19,9 X10,18,1,17 X8,20,9,19
Gauss code	{1, -5, 2, -6, 4, -7, 8, -11, 9, -10}, {5, -1, 3, -2, 6, -4, 10, -9, 11, -8, 7, -3}

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^4 v^2-u^4 v+u^3 v^3-3 u^3 v^2+2 u^3 v-u^3+u^2 v^4-3 u^2 v^3+3 u^2 v^2-3 u^2 v+u^2-u v^4+2 u v^3-3 u v^2+u v-v^3+v^2}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 9 q^{9/2}-8 q^{7/2}+6 q^{5/2}-4 q^{3/2}+q^{21/2}-2 q^{19/2}+3 q^{17/2}-6 q^{15/2}+7 q^{13/2}-9 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-3} -z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} -3 z^3 a^{-7} +z^3 a^{-9} +2 z a^{-1} +2 z a^{-5} -3 z a^{-7} +2 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-12} -4 z^4 a^{-12} +3 z^2 a^{-12} +2 z^7 a^{-11} -8 z^5 a^{-11} +8 z^3 a^{-11} -3 z a^{-11} +2 z^8 a^{-10} -6 z^6 a^{-10} +2 z^4 a^{-10} +z^2 a^{-10} +2 z^9 a^{-9} -7 z^7 a^{-9} +7 z^5 a^{-9} -z^3 a^{-9} -3 z a^{-9} +z^{10} a^{-8} -2 z^8 a^{-8} -z^6 a^{-8} +5 z^4 a^{-8} -2 z^2 a^{-8} +4 z^9 a^{-7} -18 z^7 a^{-7} +33 z^5 a^{-7} -23 z^3 a^{-7} +8 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} -2 z^8 a^{-6} +8 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} +2 z^9 a^{-5} -7 z^7 a^{-5} +14 z^5 a^{-5} -12 z^3 a^{-5} +6 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -4 z^6 a^{-4} +4 z^4 a^{-4} -3 z^2 a^{-4} +2 z^7 a^{-3} -3 z^5 a^{-3} -z^3 a^{-3} +2 z^6 a^{-2} -5 z^4 a^{-2} +2 z^2 a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +2 z a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     1   1 18                   2 1   -1 16                 4 1     3 14               4 3       -1 12             5 3         2 10           4 4           0 8         4 5             -1 6       2 4               2 4     2 4                 -2 2   1 3                   2 0   1                     -1 -2 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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