L10n15

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Link Presentations

[edit Notes on L10n15's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,20 X16,12,17,11 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, 4, -9, 8, 6, -7, -4, 9, -8, 10, -2, 3, 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-1) (v-1) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-7 q^{3/2}+\frac{1}{q^{3/2}}+2 q^{13/2}-4 q^{11/2}+4 \sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} + a^{-7} z^{-1} -z^5 a^{-5} -4 z^3 a^{-5} -5 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +5 z a^{-3} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^2 a^{-8} -2 a^{-8} +z^5 a^{-7} +3 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +4 z^6 a^{-6} -9 z^4 a^{-6} +12 z^2 a^{-6} -5 a^{-6} +5 z^7 a^{-5} -15 z^5 a^{-5} +18 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^8 a^{-4} -12 z^4 a^{-4} +12 z^2 a^{-4} -3 a^{-4} +8 z^7 a^{-3} -27 z^5 a^{-3} +26 z^3 a^{-3} -10 z a^{-3} +2 z^8 a^{-2} -3 z^6 a^{-2} -6 z^4 a^{-2} +5 z^2 a^{-2} + a^{-2} +3 z^7 a^{-1} -11 z^5 a^{-1} +11 z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} +z^6-3 z^4+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   \ -3 -2 -1 0 1 2 3 4 5 χ 14                 2 -2 12               2   2 10             4 2   -2 8           3 2     1 6         3 4       1 4       4 3         1 2     2 5           3 0   1 2             -1 -2   2               2 -4 1                 -1

Integral Khovanov Homology (db, data source)

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