L11a100

L11a99

L11a101

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Link Presentations

[edit Notes on L11a100's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X20,11,21,12 X8,17,9,18 X10,19,11,20 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X2,16,3,15
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -5, 3, -6, 4, -9, 10, -8, 11, -2, 5, -3, 6, -4, 7, -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-1) (v-1) \left(v^2-v+1\right)^2 \left(v^2+v+1\right)}{\sqrt{u} v^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^9-a^5 z^7+7 a^3 z^7-a z^7-5 a^5 z^5+17 a^3 z^5-5 a z^5-7 a^5 z^3+17 a^3 z^3-7 a z^3-3 a^5 z+7 a^3 z-4 a z-a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-a^{10} z^2+3 a^9 z^5-4 a^9 z^3+4 a^8 z^6-5 a^8 z^4+a^8 z^2+4 a^7 z^7-5 a^7 z^5+2 a^7 z^3+4 a^6 z^8-8 a^6 z^6+7 a^6 z^4-2 a^6 z^2+a^6+4 a^5 z^9-14 a^5 z^7+23 a^5 z^5-18 a^5 z^3+6 a^5 z-a^5 z^{-1} +2 a^4 z^{10}-4 a^4 z^8-3 a^4 z^6+11 a^4 z^4-8 a^4 z^2+3 a^4+8 a^3 z^9-38 a^3 z^7+63 a^3 z^5-46 a^3 z^3+13 a^3 z-3 a^3 z^{-1} +2 a^2 z^{10}-5 a^2 z^8-4 a^2 z^6+11 a^2 z^4-6 a^2 z^2+3 a^2+4 a z^9-19 a z^7+z^7 a^{-1} +28 a z^5-4 z^5 a^{-1} -19 a z^3+3 z^3 a^{-1} +7 a z-2 a z^{-1} +3 z^8-13 z^6+13 z^4-2 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 6                       1 1 4                     2   -2 2                   3 1   2 0                 3 2     -1 -2               7 3       4 -4             5 5         0 -6           6 5           1 -8         4 5             1 -10       3 6               -3 -12     2 4                 2 -14   1 3                   -2 -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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