L11a284

L11a283

L11a285

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Link Presentations

[edit Notes on L11a284's Link Presentations]

Planar diagram presentation	X10,1,11,2 X22,11,9,12 X8,9,1,10 X2,22,3,21 X14,6,15,5 X12,4,13,3 X4,14,5,13 X16,20,17,19 X18,8,19,7 X6,18,7,17 X20,16,21,15
Gauss code	{1, -4, 6, -7, 5, -10, 9, -3}, {3, -1, 2, -6, 7, -5, 11, -8, 10, -9, 8, -11, 4, -2}

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

Integral Khovanov Homology (db, data source)

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