L10a34

L10a33

L10a35

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Link Presentations

[edit Notes on L10a34's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,8,17,7 X20,18,5,17 X18,9,19,10 X8,19,9,20 X14,12,15,11 X10,16,11,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 5, -8, 7, -2, 10, -7, 8, -3, 4, -5, 6, -4}

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)-1) (t(2)-2) (t(2)-1) (2 t(2)-1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -7 q^{9/2}+9 q^{7/2}-\frac{1}{q^{7/2}}-12 q^{5/2}+\frac{2}{q^{5/2}}+12 q^{3/2}-\frac{5}{q^{3/2}}-q^{13/2}+4 q^{11/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+a^3 z^{-1} - a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3+z^3 a^{-1} -3 a z+2 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -6 z^8 a^{-2} -4 z^8 a^{-4} -2 z^8-2 a z^7-4 z^7 a^{-1} -8 z^7 a^{-3} -6 z^7 a^{-5} -2 a^2 z^6+8 z^6 a^{-2} +3 z^6 a^{-4} -4 z^6 a^{-6} -z^6-a^3 z^5-a z^5+5 z^5 a^{-1} +18 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +4 a^2 z^4-3 z^4 a^{-2} +4 z^4 a^{-4} +7 z^4 a^{-6} +4 z^4+3 a^3 z^3+9 a z^3+2 z^3 a^{-1} -10 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} -2 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} -z^2 a^{-6} -2 z^2-3 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} 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 14                     1 1 12                   3   -3 10                 4 1   3 8               5 3     -2 6             7 4       3 4           5 5         0 2         6 7           -1 0       4 7             3 -2     1 4               -3 -4   1 4                 3 -6   1                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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