L10a78

L10a77

L10a79

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Link Presentations

[edit Notes on L10a78's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X18,12,19,11 X16,5,17,6 X12,18,13,17 X20,16,7,15 X4,19,5,20
Gauss code	{1, -2, 3, -10, 7, -5}, {4, -1, 2, -3, 6, -8, 5, -4, 9, -7, 8, -6, 10, -9}

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)^4-4 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-4 t(2)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{8}{q^{9/2}}-\frac{11}{q^{7/2}}-q^{5/2}+\frac{10}{q^{5/2}}+3 q^{3/2}-\frac{10}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{11/2}}-5 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^5+3 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-10 a^3 z^3-10 a^3 z-3 a^3 z^{-1} +2 a z^5+7 a z^3-z^3 a^{-1} +6 a z-2 z a^{-1} +a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^9-3 z^4 a^8-6 z^5 a^7+5 z^3 a^7-2 z a^7-8 z^6 a^6+10 z^4 a^6-2 z^2 a^6-9 z^7 a^5+19 z^5 a^5-13 z^3 a^5+7 z a^5-2 a^5 z^{-1} -6 z^8 a^4+10 z^6 a^4+4 z^4 a^4-7 z^2 a^4+3 a^4-2 z^9 a^3-5 z^7 a^3+31 z^5 a^3-31 z^3 a^3+12 z a^3-3 a^3 z^{-1} -9 z^8 a^2+31 z^6 a^2-26 z^4 a^2+z^2 a^2+3 a^2-2 z^9 a+3 z^7 a+10 z^5 a-17 z^3 a+5 z a-a z^{-1} -3 z^8+13 z^6-17 z^4+6 z^2+1-z^7 a^{-1} +4 z^5 a^{-1} -5 z^3 a^{-1} +2 z a^{-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                   2   -2 2                 3 1   2 0               5 2     -3 -2             5 3       2 -4           6 6         0 -6         5 4           1 -8       3 6             3 -10     3 5               -2 -12     3                 3 -14 1 3                   -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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