L11a339

L11a338

L11a340

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Link Presentations

[edit Notes on L11a339's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,4,15,3 X22,5,9,6 X6,9,7,10 X18,12,19,11 X20,14,21,13 X12,20,13,19 X16,8,17,7 X4,16,5,15 X8,18,1,17 X2,21,3,22
Gauss code	{1, -11, 2, -9, 3, -4, 8, -10}, {4, -1, 5, -7, 6, -2, 9, -8, 10, -5, 7, -6, 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(1)-1) (t(2)-1) \left(t(1)^2 t(2)^4-t(1) t(2)^4+3 t(1) t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2+3 t(1) t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 14 q^{9/2}-17 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^9 a^{-3} +z^7 a^{-1} -7 z^7 a^{-3} +z^7 a^{-5} +5 z^5 a^{-1} -19 z^5 a^{-3} +5 z^5 a^{-5} +9 z^3 a^{-1} -25 z^3 a^{-3} +9 z^3 a^{-5} +8 z a^{-1} -15 z a^{-3} +7 z a^{-5} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10} a^{-4} -4 z^9 a^{-1} -10 z^9 a^{-3} -6 z^9 a^{-5} -z^8 a^{-2} -6 z^8 a^{-4} -8 z^8 a^{-6} -3 z^8-a z^7+13 z^7 a^{-1} +33 z^7 a^{-3} +11 z^7 a^{-5} -8 z^7 a^{-7} +20 z^6 a^{-2} +29 z^6 a^{-4} +14 z^6 a^{-6} -6 z^6 a^{-8} +11 z^6+4 a z^5-11 z^5 a^{-1} -41 z^5 a^{-3} -11 z^5 a^{-5} +12 z^5 a^{-7} -3 z^5 a^{-9} -24 z^4 a^{-2} -33 z^4 a^{-4} -12 z^4 a^{-6} +7 z^4 a^{-8} -z^4 a^{-10} -11 z^4-5 a z^3+6 z^3 a^{-1} +35 z^3 a^{-3} +13 z^3 a^{-5} -9 z^3 a^{-7} +2 z^3 a^{-9} +10 z^2 a^{-2} +18 z^2 a^{-4} +6 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +3 z^2+2 a z-7 z a^{-1} -17 z a^{-3} -6 z a^{-5} +2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 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                   5 1   -4 12                 6 2     4 10               8 5       -3 8             9 6         3 6           7 8           1 4         7 9             -2 2       5 9               4 0     2 5                 -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}^{2} }[/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}^{6}\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^{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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