L10a148

L10a147

L10a149

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Link Presentations

[edit Notes on L10a148's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X8,20,9,19 X18,8,19,7 X20,10,13,9 X10,14,11,13 X2,11,3,12 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {3, -1, 6, -5, 7, -8, 9, -4}, {8, -2, 4, -3, 10, -6, 5, -7}

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^3 w^3-u v^3 w^2-u v^2 w^3+3 u v^2 w^2-u v^2 w-2 u v w^2+3 u v w-u v-u w+u-v^3 w^3+v^3 w^2+v^2 w^3-3 v^2 w^2+2 v^2 w+v w^2-3 v w+v+w-1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+10 q^3+ q^{-3} -9 q^2-3 q^{-2} +10 q+5 q^{-1} -6 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +4 z^4 a^{-4} +4 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -12 z^4 a^{-2} -9 z^2 a^{-2} -2 a^{-2} z^{-2} -3 a^{-2} +z^6+4 z^4+4 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +8 z^8 a^{-2} +4 z^8 a^{-4} +4 z^8+3 a z^7-3 z^7 a^{-1} -2 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-30 z^6 a^{-2} -10 z^6 a^{-4} +4 z^6 a^{-6} -15 z^6-10 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -3 a^2 z^4+41 z^4 a^{-2} +15 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +17 z^4+6 a z^3+7 z^3 a^{-1} +7 z^3 a^{-3} +2 z^3 a^{-5} -4 z^3 a^{-7} +a^2 z^2-22 z^2 a^{-2} -10 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} -9 z^2-3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   2   -2 11                 3 1   2 9               5 3     -2 7             5 2       3 5           4 5         1 3         6 5           1 1       3 7             4 -1     2 3               -1 -3   1 3                 2 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

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