L10a66

L10a65

L10a67

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Link Presentations

[edit Notes on L10a66's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,17,7,18 X18,14,19,13 X12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8}

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)^4+t(1) t(2)^4+3 t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-3 t(1)^2 t(2)^2+3 t(1) t(2)^2-3 t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+3 t(2)+t(1)-1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^5+3 a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-8 a^3 z-2 a^3 z^{-1} +2 a z^5+8 a z^3-z^3 a^{-1} +8 a z-3 z a^{-1} +2 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^9-3 z^4 a^8+z^2 a^8-5 z^5 a^7+4 z^3 a^7-z a^7-6 z^6 a^6+7 z^4 a^6-z^2 a^6-6 z^7 a^5+11 z^5 a^5-5 z^3 a^5+2 z a^5-a^5 z^{-1} -4 z^8 a^4+7 z^6 a^4-z^9 a^3-6 z^7 a^3+29 z^5 a^3-31 z^3 a^3+12 z a^3-2 a^3 z^{-1} -6 z^8 a^2+22 z^6 a^2-22 z^4 a^2+7 z^2 a^2-a^2-z^9 a-z^7 a+18 z^5 a-29 z^3 a+14 z a-2 a z^{-1} -2 z^8+9 z^6-12 z^4+5 z^2-z^7 a^{-1} +5 z^5 a^{-1} -8 z^3 a^{-1} +5 z a^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                   1   -1 2                 3 1   2 0               3 1     -2 -2             5 3       2 -4           4 4         0 -6         5 4           1 -8       3 5             2 -10     2 4               -2 -12   1 3                 2 -14   2                   -2 -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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