L11a4

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Link Presentations

[edit Notes on L11a4's Link Presentations]

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