L10a55

L10a54

L10a56

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Link Presentations

[edit Notes on L10a55's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X16,13,17,14 X6,18,1,17 X18,12,19,11 X12,6,13,5 X20,16,7,15 X4,19,5,20
Gauss code	{1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -5, 6, -7, 10, -9}

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^4-4 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-9 u v^2+5 u v-u-v^3+4 v^2-4 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-2 a^3 z^3+z^3 a^{-3} +2 z a^{-3} +a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +a z-4 z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -6 a^2 z^8-4 z^8 a^{-2} -10 z^8-8 a^3 z^7-8 a z^7-3 z^7 a^{-1} -3 z^7 a^{-3} -7 a^4 z^6+5 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +23 z^6-4 a^5 z^5+9 a^3 z^5+23 a z^5+19 z^5 a^{-1} +9 z^5 a^{-3} -a^6 z^4+7 a^4 z^4+2 a^2 z^4-6 z^4 a^{-2} +3 z^4 a^{-4} -15 z^4+3 a^5 z^3-a^3 z^3-14 a z^3-18 z^3 a^{-1} -8 z^3 a^{-3} -a^4 z^2-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-2 a^3 z+5 z a^{-1} +3 z a^{-3} -a^2+a^3 z^{-1} +a 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 10                     1 -1 8                   2   2 6                 4 1   -3 4               6 2     4 2             6 4       -2 0           8 6         2 -2         6 7           1 -4       5 7             -2 -6     3 7               4 -8   1 4                 -3 -10   3                   3 -12 1                     -1

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=-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}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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