L11n114

L11n113

L11n115

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Link Presentations

[edit Notes on L11n114's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, 9, -7, -2, 11, 8, -9, 7, -8, -3, 4, -5, 6, -4}

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{(t(1)-1) (t(2)-1) \left(t(2)^4-2 t(2)^3+t(2)^2-2 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{2}{q^{3/2}}+2 q^{13/2}-5 q^{11/2}+7 \sqrt{q}-\frac{5}{\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} -6 z a^{-5} -4 a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +12 z a^{-3} +6 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-10 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -7 z^8 a^{-2} -5 z^8 a^{-4} -2 z^8-a z^7-3 z^7 a^{-1} -10 z^7 a^{-3} -8 z^7 a^{-5} +23 z^6 a^{-2} +10 z^6 a^{-4} -5 z^6 a^{-6} +8 z^6+5 a z^5+26 z^5 a^{-1} +47 z^5 a^{-3} +25 z^5 a^{-5} -z^5 a^{-7} -16 z^4 a^{-2} +2 z^4 a^{-4} +9 z^4 a^{-6} -9 z^4-9 a z^3-41 z^3 a^{-1} -58 z^3 a^{-3} -31 z^3 a^{-5} -5 z^3 a^{-7} -8 z^2 a^{-4} -9 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2+7 a z+24 z a^{-1} +31 z a^{-3} +18 z a^{-5} +4 z a^{-7} + a^{-2} +3 a^{-4} +3 a^{-6} + a^{-8} +1-2 a z^{-1} -5 a^{-1} z^{-1} -6 a^{-3} z^{-1} -4 a^{-5} z^{-1} - 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   2 -2 12                 3   3 10               4 2   -2 8             5 3     2 6           4 4       0 4         5 5         0 2       4 6           2 0     1 3             -2 -2   1 4               3 -4   1                 -1 -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=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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