L11n50

L11n49

L11n51

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Link Presentations

[edit Notes on L11n50's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, -4, 9, -8, 5, -6, 4, -7, 8, -9, 7}

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