L11a26

L11a25

L11a27

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Link Presentations

[edit Notes on L11a26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,5,15 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 7, -8, 4, -5, 6, -4, 8, -7}

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{2 t(1) t(2)^3-5 t(2)^3-8 t(1) t(2)^2+10 t(2)^2+10 t(1) t(2)-8 t(2)-5 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -11 q^{9/2}+14 q^{7/2}-\frac{1}{q^{7/2}}-16 q^{5/2}+\frac{2}{q^{5/2}}+16 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-3 q^{13/2}+7 q^{11/2}-14 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -2 z^3 a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} -z^3 a^{-3} +a^3 z-3 z a^{-3} +a^3 z^{-1} -2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z+z a^{-1} -a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -8 z^5 a^{-7} +5 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -14 z^6 a^{-6} +13 z^4 a^{-6} -9 z^2 a^{-6} +3 a^{-6} +4 z^9 a^{-5} -7 z^7 a^{-5} +z^5 a^{-5} -z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} +z^{10} a^{-4} +9 z^8 a^{-4} -34 z^6 a^{-4} +46 z^4 a^{-4} -30 z^2 a^{-4} +7 a^{-4} +7 z^9 a^{-3} -16 z^7 a^{-3} +a^3 z^5+20 z^5 a^{-3} -3 a^3 z^3-18 z^3 a^{-3} +3 a^3 z+10 z a^{-3} -a^3 z^{-1} -2 a^{-3} z^{-1} +z^{10} a^{-2} +7 z^8 a^{-2} +2 a^2 z^6-21 z^6 a^{-2} -3 a^2 z^4+28 z^4 a^{-2} -16 z^2 a^{-2} +a^2+4 a^{-2} +3 z^9 a^{-1} +3 a z^7-3 z^7 a^{-1} -3 a z^5+7 z^5 a^{-1} -a z^3-10 z^3 a^{-1} +2 a z+6 z a^{-1} -a z^{-1} - a^{-1} z^{-1} +3 z^8-5 z^4+3 z^2 }[/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 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 6 2     4 8               8 5       -3 6             8 6         2 4           8 8           0 2         6 8             -2 0       4 9               5 -2     2 5                 -3 -4     4                   4 -6 1 2                     -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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