L11n159

L11n158

L11n160

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Link Presentations

[edit Notes on L11n159's Link Presentations]

Planar diagram presentation	X8192 X7,17,8,16 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X21,1,22,6 X20,14,21,13 X17,7,18,22 X4,20,5,19
Gauss code	{1, -4, 3, -11, -7, 8}, {-2, -1, 5, -3, -6, 7, 9, -5, 4, 2, -10, 6, 11, -9, -8, 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^4-2 u^2 v^3+u^2 v^2-u v^4+u v^2-u+v^2-2 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-5} +z^5 a^{-3} -5 z^5 a^{-5} +z^5 a^{-7} +4 z^3 a^{-3} -6 z^3 a^{-5} +4 z^3 a^{-7} +3 z a^{-3} -2 z a^{-5} +z a^{-7} -z a^{-9} + a^{-3} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-5} -z^9 a^{-7} -2 z^8 a^{-4} -3 z^8 a^{-6} -z^8 a^{-8} -z^7 a^{-3} +3 z^7 a^{-5} +4 z^7 a^{-7} +10 z^6 a^{-4} +15 z^6 a^{-6} +5 z^6 a^{-8} +5 z^5 a^{-3} +4 z^5 a^{-5} -z^5 a^{-7} -12 z^4 a^{-4} -19 z^4 a^{-6} -7 z^4 a^{-8} -7 z^3 a^{-3} -12 z^3 a^{-5} -5 z^3 a^{-7} +2 z^2 a^{-4} +8 z^2 a^{-6} +7 z^2 a^{-8} +z^2 a^{-10} +4 z a^{-3} +6 z a^{-5} +3 z a^{-7} +z a^{-9} + a^{-4} -3 a^{-6} -5 a^{-8} -2 a^{-10} - a^{-3} z^{-1} +2 a^{-7} z^{-1} + a^{-9} 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 χ 16               2 -2 14             1 1 0 12           2 2   0 10         2 1 1   2 8       1 3       2 6     2 2 1       1 4   1 2           1 2   1             -1 0 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=-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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