L11n255

L11n254

L11n256

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Link Presentations

[edit Notes on L11n255's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, -3, 6, -5, 9, -8, 3, -7, 4, -6, 5, -4, 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{2 t(1) t(2) t(3)^3-2 t(2) t(3)^3+2 t(3)^3+t(1) t(3)^2-4 t(1) t(2) t(3)^2+2 t(2) t(3)^2-4 t(3)^2-2 t(1) t(3)+4 t(1) t(2) t(3)-t(2) t(3)+4 t(3)+2 t(1)-2 t(1) t(2)-2}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{-9} +6 q^{-8} -9 q^{-7} +11 q^{-6} -11 q^{-5} +12 q^{-4} -7 q^{-3} +6 q^{-2} -2 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+z^4 a^8+3 z^2 a^8+3 a^8 z^{-2} +5 a^8-z^6 a^6-3 z^4 a^6-4 z^2 a^6-2 a^6 z^{-2} -5 a^6-z^6 a^4-3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} -2 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 z^3 a^{11}-7 z a^{11}+2 a^{11} z^{-1} +3 z^6 a^{10}-2 z^2 a^{10}-a^{10} z^{-2} +a^{10}+7 z^7 a^9-20 z^5 a^9+35 z^3 a^9-27 z a^9+8 a^9 z^{-1} +5 z^8 a^8-10 z^6 a^8+13 z^4 a^8-8 z^2 a^8-3 a^8 z^{-2} +5 a^8+z^9 a^7+10 z^7 a^7-34 z^5 a^7+46 z^3 a^7-34 z a^7+10 a^7 z^{-1} +7 z^8 a^6-15 z^6 a^6+8 z^4 a^6-3 z^2 a^6-2 a^6 z^{-2} +4 a^6+z^9 a^5+5 z^7 a^5-19 z^5 a^5+18 z^3 a^5-10 z a^5+2 a^5 z^{-1} +2 z^8 a^4-z^6 a^4-9 z^4 a^4+9 z^2 a^4+a^4 z^{-2} -3 a^4+2 z^7 a^3-5 z^5 a^3+z^3 a^3+4 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               5 1   4 -5             5 4     -1 -7           7 2       5 -9         4 5         1 -11       7 7           0 -13     3 5             2 -15   3 6               -3 -17   3                 3 -19 3                   -3

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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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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