L11a262

L11a261

L11a263

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Link Presentations

[edit Notes on L11a262's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,6,15,5 X22,13,9,14 X16,20,17,19 X18,8,19,7 X6,18,7,17 X20,16,21,15 X4,22,5,21 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, 3, -7, 6, -11}, {10, -1, 11, -2, 4, -3, 8, -5, 7, -6, 5, -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{(t(1)-1) (t(2)-1) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -8 q^{9/2}+\frac{1}{q^{9/2}}+12 q^{7/2}-\frac{3}{q^{7/2}}-16 q^{5/2}+\frac{6}{q^{5/2}}+18 q^{3/2}-\frac{11}{q^{3/2}}-q^{13/2}+4 q^{11/2}-18 \sqrt{q}+\frac{14}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} +z^5 a^{-3} -a^3 z^3-a^3 z-z a^{-3} +a z^5+2 z^5 a^{-1} +a z^3+3 z^3 a^{-1} +a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +7 z^7 a^{-5} -12 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +7 z^8 a^{-4} +a^4 z^6-10 z^6 a^{-4} -3 a^4 z^4+4 z^4 a^{-4} +2 a^4 z^2-z^2 a^{-4} +4 z^9 a^{-3} +3 a^3 z^7+2 z^7 a^{-3} -9 a^3 z^5-13 z^5 a^{-3} +8 a^3 z^3+10 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +z^{10} a^{-2} +4 a^2 z^8+11 z^8 a^{-2} -9 a^2 z^6-27 z^6 a^{-2} +4 a^2 z^4+23 z^4 a^{-2} -5 z^2 a^{-2} +3 a z^9+7 z^9 a^{-1} -a z^7-9 z^7 a^{-1} -11 a z^5-2 z^5 a^{-1} +13 a z^3+9 z^3 a^{-1} -6 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+8 z^8-23 z^6+20 z^4-5 z^2-1 }[/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 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 7 3     -4 6               9 5       4 4             9 7         -2 2           9 9           0 0         7 11             4 -2       4 7               -3 -4     2 7                 5 -6   1 4                   -3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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