L11n36

L11n35

L11n37

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Link Presentations

[edit Notes on L11n36's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -9, 10, -4, 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{u v^5-4 u v^4+5 u v^3-3 u v^2-3 v^3+5 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-2 q^{7/2}+4 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+5 a z^5-2 z^5 a^{-1} -4 a^3 z^3+9 a z^3-8 z^3 a^{-1} +z^3 a^{-3} +a^5 z-5 a^3 z+8 a z-9 z a^{-1} +3 z a^{-3} +a^5 z^{-1} -2 a^3 z^{-1} +3 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^5 z^3-4 a^5 z+a^5 z^{-1} +a^4 z^6+z^6 a^{-4} +2 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2+4 z^2 a^{-4} - a^{-4} +3 a^3 z^7+2 z^7 a^{-3} -7 a^3 z^5-7 z^5 a^{-3} +13 a^3 z^3+7 z^3 a^{-3} -9 a^3 z-4 z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -8 a^2 z^6-4 z^6 a^{-2} +14 a^2 z^4-3 z^4 a^{-2} -9 a^2 z^2+6 z^2 a^{-2} +2 a^2-2 a^{-2} +a z^9+z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -15 a z^5-15 z^5 a^{-1} +24 a z^3+21 z^3 a^{-1} -15 a z-14 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +5 z^8-14 z^6+13 z^4-5 z^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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 1   1 6               3 1   -2 4             4 1     3 2           4 3       -1 0         5 4         1 -2       4 5           1 -4     3 4             -1 -6   1 4               3 -8 1 3                 -2 -10 2                   2

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