L11a5

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Link Presentations

[edit Notes on L11a5's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,12,5,11 X20,14,21,13 X14,20,15,19 X12,22,13,21 X18,10,19,9 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 7, -8, 11, -2, 3, -10, 8, -7, 9, -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{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -11 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+12 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-5 q^{13/2}+8 q^{11/2}-10 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} -a z^3+z^3 a^{-1} +2 z^3 a^{-5} -z^3 a^{-7} -z a^{-3} +2 z a^{-5} -z a^{-7} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-4} -2 z^{10} a^{-6} -5 z^9 a^{-3} -9 z^9 a^{-5} -4 z^9 a^{-7} -6 z^8 a^{-2} -z^8 a^{-4} +2 z^8 a^{-6} -3 z^8 a^{-8} -6 z^7 a^{-1} +12 z^7 a^{-3} +36 z^7 a^{-5} +17 z^7 a^{-7} -z^7 a^{-9} +9 z^6 a^{-2} +18 z^6 a^{-4} +16 z^6 a^{-6} +13 z^6 a^{-8} -6 z^6-4 a z^5+4 z^5 a^{-1} -13 z^5 a^{-3} -46 z^5 a^{-5} -21 z^5 a^{-7} +4 z^5 a^{-9} -a^2 z^4-4 z^4 a^{-2} -23 z^4 a^{-4} -27 z^4 a^{-6} -15 z^4 a^{-8} +6 z^4+4 a z^3+4 z^3 a^{-1} +8 z^3 a^{-3} +22 z^3 a^{-5} +11 z^3 a^{-7} -3 z^3 a^{-9} +z^2 a^{-2} +6 z^2 a^{-4} +11 z^2 a^{-6} +6 z^2 a^{-8} -2 z a^{-3} -4 z a^{-5} -2 z a^{-7} +1-a z^{-1} - a^{-1} 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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     2   -2 14                   3 1   2 12                 5 2     -3 10               6 3       3 8             7 5         -2 6           7 6           1 4         5 7             2 2       5 7               -2 0     3 7                 4 -2   1 3                   -2 -4   3                     3 -6 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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