L11a264

L11a263

L11a265

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Link Presentations

[edit Notes on L11a264's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X20,17,21,18 X8,16,1,15 X6,20,7,19 X18,8,19,7 X16,21,17,22
Gauss code	{1, -5, 2, -6, 4, -9, 10, -8}, {5, -1, 3, -2, 6, -4, 8, -11, 7, -10, 9, -7, 11, -3}

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(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-3 t(2)^3 t(1)^2+7 t(2)^2 t(1)^2-5 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-5 t(2)^2 t(1)+7 t(2) t(1)-3 t(1)+2 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+6 q^{13/2}-10 q^{11/2}+12 q^{9/2}-15 q^{7/2}+14 q^{5/2}-12 q^{3/2}+9 \sqrt{q}-\frac{6}{\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^{-3} -2 z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-5} +a z^3-6 z^3 a^{-1} +6 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} +2 a z-3 z a^{-1} +5 z a^{-3} -4 z a^{-5} +2 z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -6 z^9 a^{-3} -3 z^9 a^{-5} -5 z^8 a^{-2} -7 z^8 a^{-4} -5 z^8 a^{-6} -3 z^8-a z^7+8 z^7 a^{-1} +12 z^7 a^{-3} -3 z^7 a^{-5} -6 z^7 a^{-7} +25 z^6 a^{-2} +21 z^6 a^{-4} +3 z^6 a^{-6} -5 z^6 a^{-8} +12 z^6+4 a z^5+4 z^5 a^{-3} +18 z^5 a^{-5} +7 z^5 a^{-7} -3 z^5 a^{-9} -24 z^4 a^{-2} -12 z^4 a^{-4} +4 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} -14 z^4-5 a z^3-8 z^3 a^{-1} -13 z^3 a^{-3} -18 z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +5 z^2 a^{-2} -3 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +5 z^2+2 a z+2 z a^{-1} +6 z a^{-3} +9 z a^{-5} +2 z a^{-7} -z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 6 2     4 10               7 5       -2 8             8 5         3 6           6 7           1 4         6 8             -2 2       4 7               3 0     2 5                 -3 -2   1 4                   3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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