L10a59

L10a58

L10a60

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Link Presentations

[edit Notes on L10a59's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X16,13,17,14 X20,18,7,17 X18,12,19,11 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 6, -7, 4, -3, 8, -4, 5, -6, 7, -5}

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^2 v^3-3 u^2 v^2+3 u^2 v-u^2+u v^4-5 u v^3+7 u v^2-5 u v+u-v^4+3 v^3-3 v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-3 q^{9/2}+6 q^{7/2}-9 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{9}{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 z^5+z^5 a^{-1} -a^3 z^3+2 a z^3-2 z^3 a^{-3} -a^3 z+3 a z-3 z a^{-1} -z a^{-3} +z a^{-5} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-6 a z^7-8 z^7 a^{-1} -5 z^7 a^{-3} -a^4 z^6+6 a^2 z^6-4 z^6 a^{-2} -5 z^6 a^{-4} +8 z^6+10 a^3 z^5+23 a z^5+20 z^5 a^{-1} +4 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4+a^2 z^4+15 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +6 z^4-10 a^3 z^3-23 a z^3-17 z^3 a^{-1} -z^3 a^{-3} +3 z^3 a^{-5} -2 a^4 z^2-4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -9 z^2+3 a^3 z+11 a z+10 z a^{-1} +z a^{-3} -z a^{-5} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 4 1   -3 6               5 2     3 4             6 4       -2 2           6 5         1 0         5 7           2 -2       4 5             -1 -4     2 6               4 -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=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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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