L11a346

L11a345

L11a347

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Link Presentations

[edit Notes on L11a346's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X20,14,21,13 X22,17,11,18 X18,21,19,22 X6,16,7,15 X16,8,17,7 X14,20,15,19 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, {11, -1, 3, -8, 6, -7, 4, -5, 8, -3, 5, -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(t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2+t(2) t(1)+t(2)^2-4 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -4 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +11 z a^{-3} +6 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-7 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -6 z^9 a^{-3} -4 z^9 a^{-5} -5 z^8 a^{-2} -11 z^8 a^{-4} -8 z^8 a^{-6} -2 z^8-a z^7+z^7 a^{-1} +4 z^7 a^{-3} -7 z^7 a^{-5} -9 z^7 a^{-7} +20 z^6 a^{-2} +28 z^6 a^{-4} +9 z^6 a^{-6} -6 z^6 a^{-8} +7 z^6+5 a z^5+17 z^5 a^{-1} +31 z^5 a^{-3} +37 z^5 a^{-5} +15 z^5 a^{-7} -3 z^5 a^{-9} -13 z^4 a^{-2} -10 z^4 a^{-4} +3 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -7 z^4-9 a z^3-33 z^3 a^{-1} -48 z^3 a^{-3} -41 z^3 a^{-5} -15 z^3 a^{-7} +2 z^3 a^{-9} -z^2 a^{-2} -6 z^2 a^{-4} -9 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +z^2+7 a z+22 z a^{-1} +29 z a^{-3} +20 z a^{-5} +6 z a^{-7} + a^{-2} +3 a^{-4} +3 a^{-6} + a^{-8} +1-2 a z^{-1} -5 a^{-1} z^{-1} -6 a^{-3} z^{-1} -4 a^{-5} z^{-1} - a^{-7} 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                   5 1   -4 12                 7 2     5 10               8 5       -3 8             9 7         2 6           7 8           1 4         7 9             -2 2       5 9               4 0     1 5                 -4 -2   1 5                   4 -4   1                     -1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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