L11n170

L11n169

L11n171

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Link Presentations

[edit Notes on L11n170's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,21,14,20 X19,13,20,12 X22,16,7,15 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 9, -8, -5, 10, -7, 4, 2, -9, 8, 11, -10}

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