L11a96

L11a95

L11a97

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Link Presentations

[edit Notes on L11a96's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,8,21,7 X22,15,5,16 X16,21,17,22 X18,10,19,9 X14,12,15,11 X10,18,11,17 X8,20,9,19 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 6, -8, 7, -2, 11, -7, 4, -5, 8, -6, 9, -3, 5, -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)-1) \left(2 t(2)^2-7 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -9 q^{9/2}+12 q^{7/2}-\frac{1}{q^{7/2}}-14 q^{5/2}+\frac{2}{q^{5/2}}+14 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-3 q^{13/2}+5 q^{11/2}-12 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -2 z^3 a^{-5} -2 z 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^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -5 z^9 a^{-3} -3 z^9 a^{-5} -3 z^8 a^{-2} -4 z^8 a^{-4} -4 z^8 a^{-6} -3 z^8-3 a z^7-2 z^7 a^{-1} +9 z^7 a^{-3} +5 z^7 a^{-5} -3 z^7 a^{-7} -2 a^2 z^6+7 z^6 a^{-2} +16 z^6 a^{-4} +12 z^6 a^{-6} -z^6 a^{-8} +2 z^6-a^3 z^5+3 a z^5+8 z^5 a^{-1} -z^5 a^{-3} +5 z^5 a^{-5} +10 z^5 a^{-7} +3 a^2 z^4-9 z^4 a^{-2} -17 z^4 a^{-4} -9 z^4 a^{-6} +3 z^4 a^{-8} -z^4+3 a^3 z^3+2 a z^3-11 z^3 a^{-1} -15 z^3 a^{-3} -13 z^3 a^{-5} -8 z^3 a^{-7} +7 z^2 a^{-2} +5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-3 a^3 z-3 a z+8 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} -a^2-3 a^{-2} - a^{-4} -2+a^3 z^{-1} +a z^{-1} - a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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 7 χ 16                       1 -1 14                     2   2 12                   3 1   -2 10                 6 2     4 8               6 3       -3 6             8 6         2 4           6 6           0 2         6 8             -2 0       5 8               3 -2     1 4                 -3 -4   1 5                   4 -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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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