L10a135

L10a134

L10a136

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Link Presentations

[edit Notes on L10a135's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X16,12,17,11 X10,13,5,14 X20,18,11,17 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, {4, -2, 5, -3, 10, -4, 6, -8, 7, -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{2 u v^2 w^2-3 u v^2 w+u v^2-2 u v w^2+6 u v w-3 u v-2 u w+2 u-2 v^2 w^2+2 v^2 w+3 v w^2-6 v w+2 v-w^2+3 w-2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+7 q^4+ q^{-4} -10 q^3-3 q^{-3} +14 q^2+7 q^{-2} -13 q-10 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2+z^2 a^{-2} +z^2 a^{-4} -5 z^2+2 a^2+4 a^{-2} - a^{-4} -5+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -2 z^4 a^{-6} +z^2 a^{-6} +4 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} +z a^{-5} +5 z^8 a^{-4} -12 z^6 a^{-4} +a^4 z^4+6 z^4 a^{-4} -a^4 z^2+z^2 a^{-4} -2 a^{-4} +2 z^9 a^{-3} +5 z^7 a^{-3} +3 a^3 z^5-20 z^5 a^{-3} -2 a^3 z^3+8 z^3 a^{-3} +3 z a^{-3} +10 z^8 a^{-2} +6 a^2 z^6-18 z^6 a^{-2} -8 a^2 z^4+8 a^2 z^2+11 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-8 a^{-2} +2 z^9 a^{-1} +7 a z^7+8 z^7 a^{-1} -8 a z^5-20 z^5 a^{-1} +3 a z^3+6 z^3 a^{-1} +3 a z+5 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +5 z^8+z^6-17 z^4+20 z^2+2 z^{-2} -9 }[/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 13                     1 1 11                   3   -3 9                 4 1   3 7               6 3     -3 5             8 4       4 3           6 7         1 1         8 7           1 -1       4 8             4 -3     3 6               -3 -5   1 5                 4 -7   2                   -2 -9 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=1 }[/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}^{5}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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