L11a178

L11a177

L11a179

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Link Presentations

[edit Notes on L11a178's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X4,12,5,11 X6,18,1,17 X2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -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{2 u^2 v^4-5 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-3 u v^4+10 u v^3-15 u v^2+10 u v-3 u+v^4-4 v^3+6 v^2-5 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -16 q^{9/2}+21 q^{7/2}-\frac{1}{q^{7/2}}-25 q^{5/2}+\frac{4}{q^{5/2}}+25 q^{3/2}-\frac{10}{q^{3/2}}+q^{15/2}-4 q^{13/2}+9 q^{11/2}-22 \sqrt{q}+\frac{16}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} -z^7 a^{-3} +a z^5-3 z^5 a^{-1} -3 z^5 a^{-3} +z^5 a^{-5} +2 a z^3-4 z^3 a^{-1} -3 z^3 a^{-3} +2 z^3 a^{-5} +2 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^{10} a^{-2} -3 z^{10} a^{-4} -9 z^9 a^{-1} -16 z^9 a^{-3} -7 z^9 a^{-5} -15 z^8 a^{-2} -10 z^8 a^{-4} -7 z^8 a^{-6} -12 z^8-9 a z^7+7 z^7 a^{-1} +29 z^7 a^{-3} +9 z^7 a^{-5} -4 z^7 a^{-7} -4 a^2 z^6+42 z^6 a^{-2} +32 z^6 a^{-4} +14 z^6 a^{-6} -z^6 a^{-8} +21 z^6-a^3 z^5+14 a z^5+9 z^5 a^{-1} -12 z^5 a^{-3} +3 z^5 a^{-5} +9 z^5 a^{-7} +4 a^2 z^4-34 z^4 a^{-2} -23 z^4 a^{-4} -7 z^4 a^{-6} +2 z^4 a^{-8} -16 z^4+a^3 z^3-9 a z^3-16 z^3 a^{-1} -5 z^3 a^{-3} -5 z^3 a^{-5} -6 z^3 a^{-7} +9 z^2 a^{-2} +6 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +5 z^2+4 a z+9 z a^{-1} +7 z a^{-3} +3 z a^{-5} +z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   6 1   -5 10                 10 3     7 8               11 6       -5 6             14 10         4 4           12 12           0 2         10 13             -3 0       7 13               6 -2     3 9                 -6 -4   1 7                   6 -6   3                     -3 -8 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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