L11a21

L11a20

L11a22

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Link Presentations

[edit Notes on L11a21's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,10,19,9 X20,14,21,13 X22,16,5,15 X14,22,15,21 X16,20,17,19 X8,18,9,17 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -7, 6, -8, 9, -4, 8, -5, 7, -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{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{23/2}-4 q^{21/2}+9 q^{19/2}-13 q^{17/2}+17 q^{15/2}-18 q^{13/2}+17 q^{11/2}-15 q^{9/2}+9 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\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} -3 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -6 z^3 a^{-5} -2 z^3 a^{-7} +2 z^3 a^{-9} +5 z a^{-3} -5 z a^{-5} -z a^{-7} +z a^{-9} +2 a^{-3} z^{-1} -2 a^{-5} z^{-1} - a^{-7} z^{-1} + a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-14} +4 z^5 a^{-13} -z^3 a^{-13} +9 z^6 a^{-12} -9 z^4 a^{-12} +4 z^2 a^{-12} - a^{-12} +12 z^7 a^{-11} -16 z^5 a^{-11} +7 z^3 a^{-11} -z a^{-11} +10 z^8 a^{-10} -10 z^6 a^{-10} -4 z^4 a^{-10} +4 z^2 a^{-10} +5 z^9 a^{-9} +4 z^7 a^{-9} -22 z^5 a^{-9} +12 z^3 a^{-9} -2 z a^{-9} - a^{-9} z^{-1} +z^{10} a^{-8} +13 z^8 a^{-8} -33 z^6 a^{-8} +19 z^4 a^{-8} -6 z^2 a^{-8} +3 a^{-8} +7 z^9 a^{-7} -11 z^7 a^{-7} -8 z^5 a^{-7} +15 z^3 a^{-7} -4 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} +5 z^8 a^{-6} -21 z^6 a^{-6} +19 z^4 a^{-6} -4 z^2 a^{-6} +2 z^9 a^{-5} -2 z^7 a^{-5} -11 z^5 a^{-5} +20 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +2 z^8 a^{-4} -7 z^6 a^{-4} +6 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +9 z^3 a^{-3} -7 z a^{-3} +2 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                     3   3 20                   6 1   -5 18                 7 3     4 16               10 6       -4 14             8 7         1 12           9 10           1 10         6 8             -2 8       3 9               6 6     3 6                 -3 4   1 5                   4 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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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