L10a48

L10a47

L10a49

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Link Presentations

[edit Notes on L10a48's Link Presentations]

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