L11a224

L11a223

L11a225

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Link Presentations

[edit Notes on L11a224's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,14,21,13 X22,16,7,15 X6,18,1,17 X14,22,15,21
Gauss code	{1, -5, 6, -7, 2, -10}, {7, -1, 3, -4, 5, -6, 8, -11, 9, -2, 10, -3, 4, -8, 11, -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^6-2 u^2 v^5+3 u^2 v^4-3 u^2 v^3+2 u^2 v^2-u^2 v-u v^6+4 u v^5-6 u v^4+7 u v^3-6 u v^2+4 u v-u-v^5+2 v^4-3 v^3+3 v^2-2 v+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-15 q^{3/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-5} +5 z^5 a^{-5} +9 z^3 a^{-5} +7 z a^{-5} +2 a^{-5} z^{-1} -z^9 a^{-3} -7 z^7 a^{-3} -19 z^5 a^{-3} -25 z^3 a^{-3} -16 z a^{-3} -5 a^{-3} z^{-1} +z^7 a^{-1} +5 z^5 a^{-1} +9 z^3 a^{-1} +8 z a^{-1} +3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -2 z^3 a^{-9} +6 z^6 a^{-8} -7 z^4 a^{-8} +5 z^2 a^{-8} - a^{-8} +8 z^7 a^{-7} -11 z^5 a^{-7} +7 z^3 a^{-7} +8 z^8 a^{-6} -12 z^6 a^{-6} +6 z^4 a^{-6} +6 z^9 a^{-5} -10 z^7 a^{-5} +10 z^5 a^{-5} -15 z^3 a^{-5} +9 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +6 z^8 a^{-4} -26 z^6 a^{-4} +27 z^4 a^{-4} -18 z^2 a^{-4} +5 a^{-4} +10 z^9 a^{-3} -32 z^7 a^{-3} +40 z^5 a^{-3} -37 z^3 a^{-3} +20 z a^{-3} -5 a^{-3} z^{-1} +2 z^{10} a^{-2} +z^8 a^{-2} -19 z^6 a^{-2} +24 z^4 a^{-2} -14 z^2 a^{-2} +5 a^{-2} +4 z^9 a^{-1} +a z^7-13 z^7 a^{-1} -4 a z^5+12 z^5 a^{-1} +5 a z^3-8 z^3 a^{-1} -2 a z+9 z a^{-1} -3 a^{-1} z^{-1} +3 z^8-11 z^6+11 z^4-2 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 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 6 2     4 10               9 5       -4 8             8 6         2 6           8 9           1 4         7 8             -1 2       4 9               5 0     3 6                 -3 -2   1 5                   4 -4   2                     -2 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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