L11a290

L11a289

L11a291

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Link Presentations

[edit Notes on L11a290's Link Presentations]

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

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{u^3 v^4-3 u^3 v^3+3 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+8 u^2 v^3-9 u^2 v^2+4 u^2 v-u^2-u v^5+4 u v^4-9 u v^3+8 u v^2-4 u v+u-v^4+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{23}{q^{5/2}}+\frac{19}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+3 a z^5-z^5 a^{-1} -2 a^5 z^3+2 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^3 z+a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+4 a^7 z^7-10 a^7 z^5+5 a^7 z^3+7 a^6 z^8-18 a^6 z^6+13 a^6 z^4-3 a^6 z^2+7 a^5 z^9-15 a^5 z^7+9 a^5 z^5-a^5 z^3-2 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+6 a^4 z^8-29 a^4 z^6+30 a^4 z^4-8 a^4 z^2-a^4+15 a^3 z^9-37 a^3 z^7+36 a^3 z^5+z^5 a^{-3} -13 a^3 z^3-z^3 a^{-3} -a^3 z+a^3 z^{-1} +3 a^2 z^{10}+9 a^2 z^8-30 a^2 z^6+4 z^6 a^{-2} +29 a^2 z^4-5 z^4 a^{-2} -9 a^2 z^2+z^2 a^{-2} +8 a z^9-10 a z^7+8 z^7 a^{-1} +4 a z^5-12 z^5 a^{-1} -a z^3+5 z^3 a^{-1} -z a^{-1} +10 z^8-16 z^6+9 z^4-3 z^2 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               11 6       5 -2             12 10         -2 -4           11 10           1 -6         8 12             4 -8       6 11               -5 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/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}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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