L10a71

L10a70

L10a72

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Link Presentations

[edit Notes on L10a71's Link Presentations]

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

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^2 v^4-4 u^2 v^3+5 u^2 v^2-3 u^2 v+u^2-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u+v^4-3 v^3+5 v^2-4 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-9 q^{9/2}+14 q^{7/2}-18 q^{5/2}+19 q^{3/2}-19 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} +a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a z+2 z a^{-1} +z a^{-3} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^9 a^{-1} -3 z^9 a^{-3} -17 z^8 a^{-2} -8 z^8 a^{-4} -9 z^8-10 a z^7-17 z^7 a^{-1} -15 z^7 a^{-3} -8 z^7 a^{-5} -5 a^2 z^6+25 z^6 a^{-2} +8 z^6 a^{-4} -4 z^6 a^{-6} +8 z^6-a^3 z^5+16 a z^5+42 z^5 a^{-1} +39 z^5 a^{-3} +13 z^5 a^{-5} -z^5 a^{-7} +5 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} +5 z^4 a^{-6} +5 z^4-6 a z^3-22 z^3 a^{-1} -26 z^3 a^{-3} -9 z^3 a^{-5} +z^3 a^{-7} -2 z^2 a^{-2} -2 z^2 a^{-4} -2 z^2 a^{-6} -2 z^2-a z+4 z a^{-3} +3 z a^{-5} - a^{-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                 6 1   5 8               8 3     -5 6             10 6       4 4           10 9         -1 2         9 9           0 0       6 11             5 -2     4 8               -4 -4   1 6                 5 -6   4                   -4 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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