L10a150

L10a149

L10a151

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Link Presentations

[edit Notes on L10a150's Link Presentations]

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

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-1) (w-1) \left(2 v w^2-2 v w+v-w^2+2 w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+8 q^8-11 q^7+13 q^6-13 q^5+13 q^4-8 q^3+6 q^2-2 q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-8} +3 a^{-2} -4 a^{-4} + a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-12} +4 z^5 a^{-11} -2 z^3 a^{-11} +8 z^6 a^{-10} -9 z^4 a^{-10} +3 z^2 a^{-10} - a^{-10} +9 z^7 a^{-9} -12 z^5 a^{-9} +5 z^3 a^{-9} -z a^{-9} +5 z^8 a^{-8} +2 z^6 a^{-8} -15 z^4 a^{-8} +11 z^2 a^{-8} -3 a^{-8} +z^9 a^{-7} +13 z^7 a^{-7} -29 z^5 a^{-7} +16 z^3 a^{-7} -3 z a^{-7} +7 z^8 a^{-6} -7 z^6 a^{-6} -13 z^4 a^{-6} +18 z^2 a^{-6} + a^{-6} z^{-2} -7 a^{-6} +z^9 a^{-5} +6 z^7 a^{-5} -18 z^5 a^{-5} +11 z^3 a^{-5} +z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} -12 z^4 a^{-4} +16 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +2 z^7 a^{-3} -5 z^5 a^{-3} +2 z^3 a^{-3} +3 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -4 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} -4 a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   3   -3 17                 5 1   4 15               6 3     -3 13             7 5       2 11           8 8         0 9         5 5           0 7       3 8             5 5     3 5               -2 3   1 5                 4 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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