L11n37

L11n36

L11n38

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Link Presentations

[edit Notes on L11n37's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X2,20,3,19
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-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-3 q^{13/2}+6 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+8 q^{3/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{1}{q^{3/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z^5 a^{-3} -3 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +a z-5 z a^{-1} +9 z a^{-3} -6 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -6 z^8 a^{-4} -3 z^8 a^{-6} -2 z^7 a^{-1} -5 z^7 a^{-3} -6 z^7 a^{-5} -3 z^7 a^{-7} +9 z^6 a^{-2} +15 z^6 a^{-4} +5 z^6 a^{-6} -z^6 a^{-8} +6 z^5 a^{-1} +24 z^5 a^{-3} +27 z^5 a^{-5} +9 z^5 a^{-7} -15 z^4 a^{-2} -9 z^4 a^{-4} +6 z^4 a^{-6} +3 z^4 a^{-8} -3 z^4-a z^3-16 z^3 a^{-1} -36 z^3 a^{-3} -28 z^3 a^{-5} -7 z^3 a^{-7} +7 z^2 a^{-2} +z^2 a^{-4} -7 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2+2 a z+11 z a^{-1} +20 z a^{-3} +14 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1} }[/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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             4 2     2 8           5 4       -1 6         5 4         1 4       3 5           2 2     4 5             -1 0   1 5               4 -2   2                 -2 -4 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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