L11a305

L11a304

L11a306

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Link Presentations

[edit Notes on L11a305's Link Presentations]

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

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^3 v^3-3 u^3 v^2+u^3 v-3 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2+u v^3-5 u v^2+8 u v-3 u+v^2-3 v+2}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 14 q^{9/2}-15 q^{7/2}+12 q^{5/2}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-6 q^{15/2}+10 q^{13/2}-13 q^{11/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-7} -3 z^3 a^{-7} -2 z a^{-7} +z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +3 z a^{-3} + a^{-3} z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +3 z^2 a^{-10} +4 z^7 a^{-9} -5 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +4 z^8 a^{-8} -4 z^6 a^{-8} -z^2 a^{-8} +3 z^9 a^{-7} -3 z^7 a^{-7} +5 z^5 a^{-7} -7 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +5 z^8 a^{-6} -15 z^6 a^{-6} +19 z^4 a^{-6} -8 z^2 a^{-6} +6 z^9 a^{-5} -15 z^7 a^{-5} +17 z^5 a^{-5} -6 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +5 z^8 a^{-4} -21 z^6 a^{-4} +25 z^4 a^{-4} -7 z^2 a^{-4} - a^{-4} +3 z^9 a^{-3} -5 z^7 a^{-3} -4 z^5 a^{-3} +10 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -12 z^6 a^{-2} +9 z^4 a^{-2} -2 z^2 a^{-2} +3 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -2 z a^{-1} +z^6-3 z^4+z^2 }[/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                   4 1   3 14                 6 2     -4 12               7 4       3 10             8 7         -1 8           7 6           1 6         5 8             3 4       4 7               -3 2     2 6                 4 0   1 3                   -2 -2   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=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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