L11n195

L11n194

L11n196

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Link Presentations

[edit Notes on L11n195's Link Presentations]

Planar diagram presentation	X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X9,18,10,19 X19,10,20,11 X22,18,7,17 X14,22,15,21
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, -8, 9, 2, -3, 7, -11, 6, -4, 10, 8, -9, -7, 11, -10}

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{t(1)^2 t(2)^4-t(1) t(2)^4-2 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+3 z^3 a^7+3 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-9 z^3 a^5-6 z a^5-a^5 z^{-1} +z^5 a^3+2 z^3 a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+2 a^{10} z^2+4 a^9 z^7-7 a^9 z^5+2 a^9 z^3-a^9 z+3 a^8 z^8-3 a^8 z^6-a^8 z^4+a^7 z^9+3 a^7 z^7-5 a^7 z^5-a^7 z^3+4 a^7 z-a^7 z^{-1} +4 a^6 z^8-6 a^6 z^6+6 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-a^5 z^7+7 a^5 z^5-10 a^5 z^3+6 a^5 z-a^5 z^{-1} +a^4 z^8+2 a^4 z^4-2 a^4 z^2+4 a^3 z^5-5 a^3 z^3+a^2 z^4-a^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 3   3 -4               4 2   -2 -6             5 2     3 -8           5 4       -1 -10         5 5         0 -12       3 5           2 -14     3 5             -2 -16   1 4               3 -18   2                 -2 -20 1                   1

Integral Khovanov Homology (db, data source)

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