L11a291

L11a290

L11a292

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Link Presentations

[edit Notes on L11a291's Link Presentations]

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

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) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-2 t(1)^2 t(2)^2+2 t(1) t(2)^2-2 t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+5 q^{9/2}-10 q^{7/2}+15 q^{5/2}-21 q^{3/2}+22 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-z^5 a^{-3} -2 a^3 z^3-z^3 a^{-3} +2 z a^{-3} +a z^7+z^7 a^{-1} +3 a z^5+2 z^5 a^{-1} +3 a z^3-2 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -4 z^{10} a^{-2} -4 z^{10}-11 a z^9-19 z^9 a^{-1} -8 z^9 a^{-3} -14 a^2 z^8-2 z^8 a^{-2} -5 z^8 a^{-4} -11 z^8-12 a^3 z^7+16 a z^7+54 z^7 a^{-1} +25 z^7 a^{-3} -z^7 a^{-5} -8 a^4 z^6+23 a^2 z^6+35 z^6 a^{-2} +15 z^6 a^{-4} +51 z^6-4 a^5 z^5+14 a^3 z^5+3 a z^5-37 z^5 a^{-1} -20 z^5 a^{-3} +2 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-10 a^2 z^4-34 z^4 a^{-2} -11 z^4 a^{-4} -40 z^4+2 a^5 z^3-4 a^3 z^3-9 a z^3-z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a^4 z^2-a^2 z^2+5 z^2 a^{-2} +5 z^2+4 a z+8 z a^{-1} +4 z a^{-3} +1-a z^{-1} - a^{-1} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     4   -4 8                   6 1   5 6                 9 4     -5 4               12 6       6 2             10 9         -1 0           13 12           1 -2         9 12             3 -4       5 11               -6 -6     3 9                 6 -8   1 5                   -4 -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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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