L11n74

L11n73

L11n75

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Link Presentations

[edit Notes on L11n74's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X13,21,14,20 X15,5,16,22 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -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{-u v^5+u v^4-u v^3+u v^2-u-v^7+v^5-v^4+v^3-v^2}{\sqrt{u} v^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{q^{3/2}}+q^{17/2}-q^{15/2}+2 q^{13/2}-3 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} +2 z^3 a^{-5} +5 z a^{-5} +3 a^{-5} z^{-1} -z^7 a^{-3} -8 z^5 a^{-3} -20 z^3 a^{-3} -19 z a^{-3} -7 a^{-3} z^{-1} +z^5 a^{-1} +6 z^3 a^{-1} +10 z a^{-1} +4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} -3 z^2 a^{-10} + a^{-10} +z^5 a^{-9} -2 z^3 a^{-9} +z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +z^7 a^{-7} -4 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +z^6 a^{-6} -2 z^4 a^{-6} +3 z^2 a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +7 z a^{-5} -3 a^{-5} z^{-1} +z^8 a^{-4} -9 z^6 a^{-4} +26 z^4 a^{-4} -26 z^2 a^{-4} +7 a^{-4} +z^9 a^{-3} -10 z^7 a^{-3} +34 z^5 a^{-3} -49 z^3 a^{-3} +30 z a^{-3} -7 a^{-3} z^{-1} +z^8 a^{-2} -9 z^6 a^{-2} +25 z^4 a^{-2} -25 z^2 a^{-2} +7 a^{-2} +z^9 a^{-1} -9 z^7 a^{-1} +28 z^5 a^{-1} -37 z^3 a^{-1} +21 z a^{-1} -4 a^{-1} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                         0 14                   2 1   -1 12                 1       1 10               2 2       0 8           1 1 1         -1 6           1 2           1 4       1 2 1             0 2         2               2 0     1                   1 -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=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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