L11a64

L11a63

L11a65

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Link Presentations

[edit Notes on L11a64's Link Presentations]

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

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)^4-2 t(2)^3+3 t(2)^2-2 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}+q^{23/2}-3 q^{21/2}+7 q^{19/2}-10 q^{17/2}+13 q^{15/2}-14 q^{13/2}+13 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -4 z^3 a^{-5} -6 z^3 a^{-7} +3 z^3 a^{-9} +4 z a^{-3} -7 z a^{-7} +3 z a^{-9} + a^{-3} z^{-1} + a^{-5} z^{-1} -4 a^{-7} z^{-1} +2 a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -2 z^3 a^{-13} +6 z^6 a^{-12} -8 z^4 a^{-12} +6 z^2 a^{-12} -2 a^{-12} +7 z^7 a^{-11} -9 z^5 a^{-11} +4 z^3 a^{-11} +z a^{-11} +6 z^8 a^{-10} -7 z^6 a^{-10} +4 z^2 a^{-10} - a^{-10} +4 z^9 a^{-9} -6 z^7 a^{-9} +9 z^5 a^{-9} -15 z^3 a^{-9} +8 z a^{-9} -2 a^{-9} z^{-1} +z^{10} a^{-8} +6 z^8 a^{-8} -21 z^6 a^{-8} +24 z^4 a^{-8} -19 z^2 a^{-8} +6 a^{-8} +6 z^9 a^{-7} -19 z^7 a^{-7} +25 z^5 a^{-7} -24 z^3 a^{-7} +13 z a^{-7} -4 a^{-7} z^{-1} +z^{10} a^{-6} +2 z^8 a^{-6} -16 z^6 a^{-6} +23 z^4 a^{-6} -16 z^2 a^{-6} +5 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -z^5 a^{-5} +5 z^3 a^{-5} +z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     2   2 20                   5 1   -4 18                 5 2     3 16               8 5       -3 14             6 5         1 12           7 8           1 10         5 6             -1 8       2 7               5 6     3 5                 -2 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=9 }[/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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