L11n368

L11n367

L11n369

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Link Presentations

[edit Notes on L11n368's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8}

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) (t(3)-1)^2 (t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1- q^{-1} +3 q^{-2} - q^{-3} + q^{-4} +2 q^{-7} -2 q^{-8} +2 q^{-9} - q^{-10} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+2 a^8 z^2+2 a^8+a^6 z^2+a^6 z^{-2} +a^6-a^4 z^6-5 a^4 z^4-7 a^4 z^2-2 a^4 z^{-2} -6 a^4+a^2 z^4+4 a^2 z^2+a^2 z^{-2} +4 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+16 a^{10} z^4-9 a^{10} z^2+2 a^{10}+a^9 z^9-4 a^9 z^7-2 a^9 z^5+11 a^9 z^3-4 a^9 z+3 a^8 z^8-20 a^8 z^6+36 a^8 z^4-22 a^8 z^2+5 a^8+a^7 z^9-6 a^7 z^7+7 a^7 z^5-a^7 z^3+a^6 z^8-7 a^6 z^6+11 a^6 z^4-6 a^6 z^2+a^6 z^{-2} -a^6+a^5 z^5-8 a^5 z^3+8 a^5 z-2 a^5 z^{-1} +3 a^4 z^6-14 a^4 z^4+15 a^4 z^2+2 a^4 z^{-2} -8 a^4+a^3 z^7-3 a^3 z^5-2 a^3 z^3+6 a^3 z-2 a^3 z^{-1} +a^2 z^6-5 a^2 z^4+8 a^2 z^2+a^2 z^{-2} -5 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                         0 -3                   3 1   2 -5               1 1 2     2 -7             1 2 1       0 -9           2 2 1         1 -11         2 4 2           0 -13       1 1 2             2 -15     1 2 1               0 -17   1 1                   0 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/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} }[/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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_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}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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