L11n185

L11n184

L11n186

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Link Presentations

[edit Notes on L11n185's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X21,1,22,6 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, 11, 2, -3, -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{2 t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+4 q^{5/2}-6 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^3+z^3 a^{-3} +a^3 z+2 z a^{-3} + a^{-3} z^{-1} -a z^5-z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} -4 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^3-a^5 z+z^6 a^{-4} +3 a^4 z^4-4 z^4 a^{-4} -3 a^4 z^2+4 z^2 a^{-4} - a^{-4} +a^3 z^7+2 z^7 a^{-3} -a^3 z^5-7 z^5 a^{-3} +3 a^3 z^3+6 z^3 a^{-3} -a^3 z-3 z a^{-3} + a^{-3} z^{-1} +2 a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-5 z^6 a^{-2} +13 a^2 z^4-z^4 a^{-2} -6 a^2 z^2+5 z^2 a^{-2} -3 a^{-2} +a z^9+z^9 a^{-1} -a z^7-2 a z^5-8 z^5 a^{-1} +6 a z^3+10 z^3 a^{-1} -4 a z-7 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +4 z^8-13 z^6+13 z^4-2 z^2-3 }[/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 χ 10                   1 -1 8                 1   1 6               3 1   -2 4             3 1     2 2           3 3       0 0         5 3         2 -2       2 4           2 -4     3 4             -1 -6   1 3               2 -8   2                 -2 -10 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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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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