L10a69

L10a68

L10a70

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Link Presentations

[edit Notes on L10a69's Link Presentations]

Planar diagram presentation	X8192 X16,9,17,10 X6718 X20,17,7,18 X10,4,11,3 X14,6,15,5 X4,12,5,11 X18,14,19,13 X12,20,13,19 X2,16,3,15
Gauss code	{1, -10, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 8, -6, 10, -2, 4, -8, 9, -4}

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{\left(v^2-3 v+1\right) (u v-u+1) (u v-v+1)}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+3 q^{11/2}-7 q^{9/2}+11 q^{7/2}-14 q^{5/2}+15 q^{3/2}-15 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +6 z^3 a^{-3} +6 z a^{-3} +2 a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +a z-4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -2 z^3 a^{-7} +z a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +5 z^7 a^{-5} -8 z^5 a^{-5} +5 z^3 a^{-5} -3 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -6 z^6 a^{-4} +3 z^4 a^{-4} -2 z^2 a^{-4} +2 z^9 a^{-3} +8 z^7 a^{-3} +a^3 z^5-23 z^5 a^{-3} -a^3 z^3+23 z^3 a^{-3} -11 z a^{-3} +2 a^{-3} z^{-1} +11 z^8 a^{-2} +4 a^2 z^6-20 z^6 a^{-2} -6 a^2 z^4+14 z^4 a^{-2} +a^2 z^2-5 z^2 a^{-2} + a^{-2} +2 z^9 a^{-1} +7 a z^7+10 z^7 a^{-1} -13 a z^5-28 z^5 a^{-1} +7 a z^3+24 z^3 a^{-1} -2 a z-9 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +6 z^8-7 z^6 }[/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 6 χ 14                     1 1 12                   2   -2 10                 5 1   4 8               6 2     -4 6             8 5       3 4           8 7         -1 2         7 7           0 0       5 9             4 -2     3 6               -3 -4   1 5                 4 -6   3                   -3 -8 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=-4 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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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