L11a50

L11a49

L11a51

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Link Presentations

[edit Notes on L11a50's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 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{2 (u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -7 q^{9/2}+\frac{1}{q^{9/2}}+11 q^{7/2}-\frac{4}{q^{7/2}}-15 q^{5/2}+\frac{7}{q^{5/2}}+18 q^{3/2}-\frac{12}{q^{3/2}}-q^{13/2}+3 q^{11/2}-18 \sqrt{q}+\frac{15}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} -a^3 z^3+z^3 a^{-3} +2 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +a z^5+2 z^5 a^{-1} +3 z^3 a^{-1} -2 a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +6 z^7 a^{-5} -14 z^5 a^{-5} +12 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +7 z^8 a^{-4} +a^4 z^6-17 z^6 a^{-4} -2 a^4 z^4+19 z^4 a^{-4} -8 z^2 a^{-4} + a^{-4} +5 z^9 a^{-3} +4 a^3 z^7-7 z^7 a^{-3} -11 a^3 z^5-z^5 a^{-3} +6 a^3 z^3+15 z^3 a^{-3} +a^3 z-11 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+5 z^8 a^{-2} -16 a^2 z^6-22 z^6 a^{-2} +10 a^2 z^4+31 z^4 a^{-2} -2 a^2 z^2-15 z^2 a^{-2} +a^2+3 a^{-2} +5 a z^9+10 z^9 a^{-1} -10 a z^7-27 z^7 a^{-1} +4 a z^5+29 z^5 a^{-1} -3 a z^3-8 z^3 a^{-1} +3 a z-4 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +2 z^{10}+4 z^8-19 z^6+19 z^4-9 z^2+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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   5 1   4 8                 6 2     -4 6               9 5       4 4             9 6         -3 2           9 9           0 0         8 11             3 -2       4 7               -3 -4     3 8                 5 -6   1 4                   -3 -8   3                     3 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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