L10a22

L10a21

L10a23

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Link Presentations

[edit Notes on L10a22's Link Presentations]

Planar diagram presentation	X6172 X14,9,15,10 X4,19,1,20 X12,6,13,5 X10,4,11,3 X20,12,5,11 X2,14,3,13 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code	{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -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)^3 \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-4 q^{15/2}+9 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 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^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +6 z^3 a^{-3} +3 z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -10 z^4 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +11 z^7 a^{-7} -15 z^5 a^{-7} +8 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +7 z^8 a^{-6} +z^6 a^{-6} -19 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +2 z^9 a^{-5} +16 z^7 a^{-5} -41 z^5 a^{-5} +26 z^3 a^{-5} -7 z a^{-5} +2 a^{-5} z^{-1} +12 z^8 a^{-4} -19 z^6 a^{-4} -4 z^4 a^{-4} +11 z^2 a^{-4} -3 a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} -33 z^5 a^{-3} +26 z^3 a^{-3} -6 z a^{-3} +5 z^8 a^{-2} -10 z^6 a^{-2} +2 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +9 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 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 χ 18                     1 -1 16                   3   3 14                 6 1   -5 12               6 3     3 10             9 6       -3 8           8 6         2 6         6 9           3 4       6 8             -2 2     3 8               5 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}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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