L11n75

L11n74

L11n76

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Link Presentations

[edit Notes on L11n75's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X15,5,16,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6}

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{2 t(2)^5-t(2)^4+2 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+2 t(2)^2-t(1) t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{9/2}+q^{7/2}-q^{5/2}-\frac{1}{q^{5/2}}+\frac{1}{q^{3/2}}+q^{15/2}-q^{13/2}+2 q^{11/2}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} +3 z a^{-5} +2 a^{-5} z^{-1} -z^5 a^{-3} -6 z^3 a^{-3} -8 z a^{-3} -4 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +3 a z-z a^{-1} +a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -z^8 a^{-4} -z^8-a z^7+6 z^7 a^{-1} +8 z^7 a^{-3} -z^7 a^{-7} +14 z^6 a^{-2} +8 z^6 a^{-4} -2 z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-9 z^5 a^{-1} -21 z^5 a^{-3} -2 z^5 a^{-5} +4 z^5 a^{-7} -27 z^4 a^{-2} -20 z^4 a^{-4} +8 z^4 a^{-6} +5 z^4 a^{-8} -4 z^4-10 a z^3+5 z^3 a^{-1} +27 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} +18 z^2 a^{-2} +21 z^2 a^{-4} -6 z^2 a^{-6} -7 z^2 a^{-8} -2 z^2+6 a z-4 z a^{-1} -17 z a^{-3} -7 z a^{-5} -5 a^{-2} -6 a^{-4} + a^{-6} +2 a^{-8} +1-a z^{-1} + a^{-1} z^{-1} +4 a^{-3} z^{-1} +2 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                         0 12                   2 1   -1 10               1 1       0 8               1 2       1 6           2 3 1         0 4         1 1 1           1 2       1 3 2             0 0     1 1 2               2 -2     1                   1 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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