L11n34

L11n33

L11n35

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Link Presentations

[edit Notes on L11n34's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,22 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, 4, -10, 9, 6, -7, -4, 8, -9, 10, -8, 11, -2, 3, 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{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{4}{q^{7/2}}-11 q^{5/2}+\frac{7}{q^{5/2}}+14 q^{3/2}-\frac{12}{q^{3/2}}-15 \sqrt{q}+\frac{14}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -a^3 z+4 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1} -a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +a^4 z^6+3 z^6 a^{-4} -2 a^4 z^4+3 z^4 a^{-4} +a^4 z^2-3 z^2 a^{-4} + a^{-4} +4 a^3 z^7+8 z^7 a^{-3} -11 a^3 z^5-16 z^5 a^{-3} +9 a^3 z^3+24 z^3 a^{-3} -a^3 z-15 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +5 a^2 z^8+7 z^8 a^{-2} -10 a^2 z^6-10 z^6 a^{-2} +2 a^2 z^4+8 z^4 a^{-2} +a^2 z^2-6 z^2 a^{-2} +a^2+3 a^{-2} +2 a z^9+2 z^9 a^{-1} +9 a z^7+13 z^7 a^{-1} -35 a z^5-40 z^5 a^{-1} +27 a z^3+36 z^3 a^{-1} -5 a z-14 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +12 z^8-24 z^6+9 z^4-3 z^2+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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                   3 3 8                 4   -4 6               7 3   4 4             7 4     -3 2           8 7       1 0         8 9         1 -2       4 6           -2 -4     3 8             5 -6   1 4               -3 -8   3                 3 -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=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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