L11a450

L11a449

L11a451

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Link Presentations

[edit Notes on L11a450's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X20,18,21,17 X16,8,17,7 X10,14,11,13 X22,12,13,11 X12,22,5,21 X8,19,9,20 X18,9,19,10 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -8, 9, -5, 6, -7}, {5, -2, 11, -4, 3, -9, 8, -3, 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{(v+w-1) (v w-v-w) (u v w-u w+u-v w+v-1)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+11 q^5-15 q^4+18 q^3+ q^{-3} -16 q^2-3 q^{-2} +16 q+7 q^{-1} -11 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-2 z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -3 z^2+a^2-3 a^{-2} +3 a^{-4} - a^{-6} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +5 z^7 a^{-7} -10 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +6 z^8 a^{-6} -13 z^6 a^{-6} +15 z^4 a^{-6} -8 z^2 a^{-6} +3 a^{-6} +4 z^9 a^{-5} -2 z^7 a^{-5} -9 z^5 a^{-5} +19 z^3 a^{-5} -7 z a^{-5} +z^{10} a^{-4} +11 z^8 a^{-4} -36 z^6 a^{-4} +52 z^4 a^{-4} -36 z^2 a^{-4} - a^{-4} z^{-2} +11 a^{-4} +7 z^9 a^{-3} -9 z^7 a^{-3} -6 z^5 a^{-3} +19 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +9 z^8 a^{-2} +a^2 z^6-28 z^6 a^{-2} -3 a^2 z^4+34 z^4 a^{-2} +3 a^2 z^2-29 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+11 a^{-2} +3 z^9 a^{-1} +3 a z^7+z^7 a^{-1} -8 a z^5-16 z^5 a^{-1} +6 a z^3+15 z^3 a^{-1} -a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-7 z^6- z^{-2} +3 }[/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 7 χ 17                       1 -1 15                     2   2 13                   4 1   -3 11                 7 2     5 9               9 5       -4 7             9 6         3 5           7 9           2 3         9 9             0 1       5 10               5 -1     2 6                 -4 -3   1 5                   4 -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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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