L11n95

L11n94

L11n96

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Link Presentations

[edit Notes on L11n95's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X19,22,20,5 X13,20,14,21 X21,14,22,15 X15,10,16,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 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{t(1) t(2)^5+t(2)^5-2 t(2)^4-t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +2 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} +2 z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-9 z a^7-2 a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-4 a^{16} z^4+3 a^{16} z^2-a^{16}+2 a^{15} z^7-8 a^{15} z^5+6 a^{15} z^3-a^{15} z+2 a^{14} z^8-8 a^{14} z^6+6 a^{14} z^4+a^{13} z^9-3 a^{13} z^7-a^{13} z^5+3 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-16 a^{12} z^6+26 a^{12} z^4-15 a^{12} z^2+3 a^{12}+a^{11} z^9-6 a^{11} z^7+12 a^{11} z^5-10 a^{11} z^3+4 a^{11} z-a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+14 a^{10} z^4-7 a^{10} z^2-2 a^9 z^5+7 a^9 z^3-7 a^9 z+2 a^9 z^{-1} -2 a^8 z^4+5 a^8 z^2-3 a^8+a^7 z^7-7 a^7 z^5+14 a^7 z^3-9 a^7 z+2 a^7 z^{-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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                       1 1 -10                 1 1     0 -12               2         2 -14             2 3 1       0 -16           3 1           2 -18         2 3 1           0 -20       2 3               -1 -22     1 2 1               0 -24   1 2                   -1 -26   1                     1 -28 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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