L11n42

L11n41

L11n43

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Link Presentations

[edit Notes on L11n42's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,22,16,5 X17,20,18,21 X21,16,22,17 X19,10,20,11 X11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 8, -9, 7}

A Braid Representative	{{{braid_table}}}
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{3 t(1) t(2)^3-4 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-4 t(2)+3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-3 z^3 a^9-2 z a^9-a^9 z^{-1} -z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^{14} z^4-8 a^{14} z^2+3 a^{14}+a^{13} z^7-a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-10 a^{12} z^6+24 a^{12} z^4-24 a^{12} z^2+7 a^{12}+a^{11} z^9-4 a^{11} z^7+10 a^{11} z^5-11 a^{11} z^3+8 a^{11} z-2 a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+25 a^{10} z^4-16 a^{10} z^2+4 a^{10}+a^9 z^9-4 a^9 z^7+9 a^9 z^5-7 a^9 z^3+4 a^9 z-a^9 z^{-1} +a^8 z^8-2 a^8 z^6+2 a^8 z^4-a^8 z^2+a^7 z^7-a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 1 1 0 -8               2     2 -10             2 1     -1 -12           4 2       2 -14         3 3         0 -16       3 3           0 -18     1 3             2 -20   2 3               -1 -22   1                 1 -24 2                   -2

Integral Khovanov Homology (db, data source)

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