L11a127

L11a126

L11a128

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Link Presentations

[edit Notes on L11a127's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,10,5,9 X18,7,19,8 X8,17,9,18 X12,19,13,20 X20,11,21,12 X10,16,11,15 X16,22,17,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 3, -8, 7, -6, 11, -2, 8, -9, 5, -4, 6, -7, 9, -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{3 u v^4-10 u v^3+14 u v^2-7 u v+u+v^5-7 v^4+14 v^3-10 v^2+3 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{19}{q^{9/2}}+\frac{22}{q^{7/2}}+q^{5/2}-\frac{23}{q^{5/2}}-4 q^{3/2}+\frac{20}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{14}{q^{11/2}}+9 \sqrt{q}-\frac{16}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -4 z a^7-3 a^7 z^{-1} +6 z^3 a^5+9 z a^5+4 a^5 z^{-1} -3 z^5 a^3-7 z^3 a^3-8 z a^3-2 a^3 z^{-1} -z^5 a+z^3 a+z a+z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-4 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-10 a^8 z^6+12 a^8 z^4-6 a^8 z^2+a^8+4 a^7 z^9-7 a^7 z^7-8 a^7 z^5+23 a^7 z^3-15 a^7 z+3 a^7 z^{-1} +2 a^6 z^{10}+9 a^6 z^8-42 a^6 z^6+44 a^6 z^4-17 a^6 z^2+3 a^6+13 a^5 z^9-22 a^5 z^7-19 a^5 z^5+44 a^5 z^3-24 a^5 z+4 a^5 z^{-1} +2 a^4 z^{10}+22 a^4 z^8-68 a^4 z^6+53 a^4 z^4-15 a^4 z^2+2 a^4+9 a^3 z^9+a^3 z^7-41 a^3 z^5+42 a^3 z^3-16 a^3 z+2 a^3 z^{-1} +16 a^2 z^8-27 a^2 z^6+14 a^2 z^4+z^4 a^{-2} -3 a^2 z^2+a^2+15 a z^7-22 a z^5+4 z^5 a^{-1} +14 a z^3-z^3 a^{-1} -3 a z+9 z^6-6 z^4+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   6 1   -5 0                 10 3     7 -2               11 7       -4 -4             12 9         3 -6           10 11           1 -8         9 12             -3 -10       6 11               5 -12     2 8                 -6 -14   1 6                   5 -16   2                     -2 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/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}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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