L11a215

L11a214

L11a216

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Link Presentations

[edit Notes on L11a215's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,12,7,11 X20,16,21,15 X18,10,19,9 X10,20,11,19 X14,22,15,21 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}

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^2 v^4-4 u^2 v^3+5 u^2 v^2-4 u^2 v+u^2-3 u v^4+6 u v^3-7 u v^2+6 u v-3 u+v^4-4 v^3+5 v^2-4 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -16 q^{9/2}+11 q^{7/2}-8 q^{5/2}+3 q^{3/2}+q^{23/2}-3 q^{21/2}+7 q^{19/2}-12 q^{17/2}+16 q^{15/2}-18 q^{13/2}+18 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} -3 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +3 z^3 a^{-3} -z^3 a^{-5} -7 z^3 a^{-7} +3 z^3 a^{-9} +3 z a^{-3} +2 z a^{-5} -7 z a^{-7} +3 z a^{-9} + a^{-3} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-6} -2 z^{10} a^{-8} -4 z^9 a^{-5} -11 z^9 a^{-7} -7 z^9 a^{-9} -3 z^8 a^{-4} -4 z^8 a^{-6} -11 z^8 a^{-8} -10 z^8 a^{-10} -z^7 a^{-3} +10 z^7 a^{-5} +30 z^7 a^{-7} +10 z^7 a^{-9} -9 z^7 a^{-11} +10 z^6 a^{-4} +29 z^6 a^{-6} +45 z^6 a^{-8} +20 z^6 a^{-10} -6 z^6 a^{-12} +4 z^5 a^{-3} -z^5 a^{-5} -22 z^5 a^{-7} +14 z^5 a^{-11} -3 z^5 a^{-13} -9 z^4 a^{-4} -33 z^4 a^{-6} -50 z^4 a^{-8} -19 z^4 a^{-10} +6 z^4 a^{-12} -z^4 a^{-14} -6 z^3 a^{-3} -6 z^3 a^{-5} +11 z^3 a^{-7} -2 z^3 a^{-9} -11 z^3 a^{-11} +2 z^3 a^{-13} +z^2 a^{-4} +15 z^2 a^{-6} +27 z^2 a^{-8} +9 z^2 a^{-10} -3 z^2 a^{-12} +z^2 a^{-14} +4 z a^{-3} -7 z a^{-7} -z a^{-9} +2 z a^{-11} + a^{-4} -3 a^{-6} -5 a^{-8} -2 a^{-10} - a^{-3} z^{-1} +2 a^{-7} z^{-1} + a^{-9} 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                 7 2     5 16               9 5       -4 14             9 7         2 12           9 9           0 10         7 9             -2 8       5 10               5 6     3 6                 -3 4   1 6                   5 2   2                     -2 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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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