L10a134

L10a133

L10a135

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Link Presentations

[edit Notes on L10a134's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,7,17,8 X20,9,11,10 X18,12,19,11 X8,15,9,16 X10,19,5,20 X14,18,15,17 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7}, {5, -2, 10, -8, 6, -3, 8, -5, 7, -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{u v^2 w^2-2 u v^2 w+u v^2-3 u v w^2+5 u v w-2 u v+3 u w^2-3 u w+u-v^2 w^2+3 v^2 w-3 v^2+2 v w^2-5 v w+3 v-w^2+2 w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -2 q^{-5} +q^4+7 q^{-4} -4 q^3-9 q^{-3} +8 q^2+13 q^{-2} -11 q-14 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -4 a^4+3 a^2 z^4+z^4 a^{-2} +6 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +4 a^2+ a^{-2} -z^6-3 z^4-5 z^2-2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-4 a^5 z^5+4 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-5 a^4 z^6-a^4 z^4+z^4 a^{-4} +6 a^4 z^2+2 a^4 z^{-2} -6 a^4+a^3 z^9+9 a^3 z^7-28 a^3 z^5+4 z^5 a^{-3} +20 a^3 z^3-2 z^3 a^{-3} -2 a^3 z^{-1} +8 a^2 z^8-9 a^2 z^6+8 z^6 a^{-2} -9 a^2 z^4-9 z^4 a^{-2} +9 a^2 z^2+4 z^2 a^{-2} +a^2 z^{-2} -3 a^2- a^{-2} +a z^9+16 a z^7+9 z^7 a^{-1} -39 a z^5-11 z^5 a^{-1} +27 a z^3+5 z^3 a^{-1} -6 a z-2 z a^{-1} +5 z^8+5 z^6-22 z^4+13 z^2-1 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   3   -3 5                 5 1   4 3               6 3     -3 1             8 5       3 -1           8 8         0 -3         5 6           -1 -5       5 9             4 -7     2 4               -2 -9     5                 5 -11 1 2                   -1 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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