L11a283

L11a282

L11a284

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Link Presentations

[edit Notes on L11a283's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,11,21,12 X8,9,1,10 X22,17,9,18 X12,4,13,3 X18,8,19,7 X6,14,7,13 X14,6,15,5 X4,16,5,15 X16,21,17,22 X2,20,3,19
Gauss code	{1, -11, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -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)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+7 t(1)^2 t(2)^3-7 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-7 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{9/2}-\frac{4}{q^{9/2}}-7 q^{7/2}+\frac{9}{q^{7/2}}+11 q^{5/2}-\frac{14}{q^{5/2}}-17 q^{3/2}+\frac{18}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+19 \sqrt{q}-\frac{20}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+4 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+2 a z^3+7 z^3 a^{-1} -3 z^3 a^{-3} -a^3 z-a z+7 z a^{-1} -3 z a^{-3} -a z^{-1} +3 a^{-1} z^{-1} -2 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4+z^7 a^{-5} +4 a^5 z^5-4 z^5 a^{-5} -a^5 z^3+5 z^3 a^{-5} -2 z a^{-5} +3 z^8 a^{-4} +9 a^4 z^6-11 z^6 a^{-4} -8 a^4 z^4+12 z^4 a^{-4} +3 a^4 z^2-3 z^2 a^{-4} +4 z^9 a^{-3} +13 a^3 z^7-12 z^7 a^{-3} -18 a^3 z^5+10 z^5 a^{-3} +9 a^3 z^3-4 z^3 a^{-3} -a^3 z+5 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +12 a^2 z^8+3 z^8 a^{-2} -16 a^2 z^6-23 z^6 a^{-2} +4 a^2 z^4+26 z^4 a^{-2} -a^2 z^2-11 z^2 a^{-2} +a^2+3 a^{-2} +7 a z^9+11 z^9 a^{-1} -2 a z^7-28 z^7 a^{-1} -14 a z^5+22 z^5 a^{-1} +4 a z^3-15 z^3 a^{-1} +2 a z+10 z a^{-1} -a z^{-1} -3 a^{-1} z^{-1} +2 z^{10}+12 z^8-37 z^6+27 z^4-12 z^2+3 }[/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 12                       1 1 10                     2   -2 8                   5 1   4 6                 7 3     -4 4               10 4       6 2             9 7         -2 0           11 10           1 -2         8 10             2 -4       6 10               -4 -6     3 8                 5 -8   1 6                   -5 -10   3                     3 -12 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=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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