L11n452

L11n451

L11n453

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Link Presentations

[edit Notes on L11n452's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X8,16,5,15 X14,8,15,7 X19,17,20,16 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {-9, 3, -5, 4}, {11, -2, -3, 6, -4, -8, 7, 9, -6, 5}

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(3)-1) (t(4)-1)^2 (t(2) t(4)-t(1))}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}+\frac{2}{q^{9/2}}-q^{7/2}-\frac{6}{q^{7/2}}-q^{5/2}+\frac{5}{q^{5/2}}+2 q^{3/2}-\frac{7}{q^{3/2}}-\frac{1}{q^{11/2}}-6 \sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-3} +a^5 z+2 a^5 z^{-1} -3 a^3 z^3-3 a^3 z^{-3} +z^3 a^{-3} -8 a^3 z-8 a^3 z^{-1} +3 z a^{-3} + a^{-3} z^{-1} +2 a z^5-z^5 a^{-1} +9 a z^3+3 a z^{-3} -7 z^3 a^{-1} - a^{-1} z^{-3} +15 a z+11 a z^{-1} -11 z a^{-1} -6 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+10 a^5 z^3-a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+z^6 a^{-4} +3 a^4 z^4-5 z^4 a^{-4} +7 a^4 z^2+4 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +a^3 z^9+3 a^3 z^7+z^7 a^{-3} -29 a^3 z^5-7 z^5 a^{-3} +45 a^3 z^3+11 z^3 a^{-3} -3 a^3 z^{-3} -34 a^3 z-9 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +6 a^2 z^8-23 a^2 z^6+12 a^2 z^4-7 z^4 a^{-2} +19 a^2 z^2+15 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +a z^9+5 a z^7+4 z^7 a^{-1} -44 a z^5-27 z^5 a^{-1} +75 a z^3+51 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -50 a z-35 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +4 z^8-17 z^6+7 z^4+23 z^2+3 z^{-2} -18 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                         0 6                 2 1 1   2 4               2 1       -1 2             5 2 1       4 0           5 8 1         2 -2         2 3 4           3 -4       3 5               2 -6     3 2                 1 -8   1 5                   4 -10   1                     -1 -12 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=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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