L10a47

L10a46

L10a48

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Link Presentations

[edit Notes on L10a47's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,8,19,7 X20,10,5,9 X8,20,9,19 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 6, -7, 10, -2, 7, -6, 8, -3, 5, -4}

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 v^3-4 u v^2+4 u v-3 u-3 v^3+4 v^2-4 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -5 q^{9/2}+7 q^{7/2}-\frac{1}{q^{7/2}}-8 q^{5/2}+\frac{1}{q^{5/2}}+9 q^{3/2}-\frac{4}{q^{3/2}}-q^{13/2}+3 q^{11/2}-8 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-1} +z^5 a^{-3} -2 a z^3+2 z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} +a^3 z-5 a z+2 z a^{-1} +z a^{-3} -z a^{-5} +2 a^3 z^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -3 z^8 a^{-4} -z^8-a z^7+2 z^7 a^{-1} -z^7 a^{-3} -4 z^7 a^{-5} -a^2 z^6+11 z^6 a^{-2} +7 z^6 a^{-4} -3 z^6 a^{-6} -a^3 z^5-a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} +a^2 z^4-16 z^4 a^{-2} -7 z^4 a^{-4} +7 z^4 a^{-6} -z^4+4 a^3 z^3+7 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} -6 z^3 a^{-5} +2 z^3 a^{-7} +3 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2-5 a^3 z-7 a z-3 z a^{-1} +z a^{-5} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 3 1   2 8               4 2     -2 6             4 3       1 4           5 4         -1 2         3 4           -1 0       3 6             3 -2     1 2               -1 -4     3                 3 -6 1 1                   0 -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{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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