L11n278

L11n277

L11n279

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Link Presentations

[edit Notes on L11n278's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, -4, -5, 10, 9, -7, -6, 11, 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{u v^2 w^2-u v^2 w+u v-v w^2+w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^4+q^3-q^2+q+ q^{-1} + q^{-2} + q^{-3} + q^{-4} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6-a^2 z^4-z^4 a^{-2} +6 z^4-6 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-9 a^2-3 a^{-2} +10+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} -a^2 z^6-6 z^6 a^{-2} -7 z^6-a^3 z^5-8 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} +a^4 z^4+7 a^2 z^4+10 z^4 a^{-2} +16 z^4+6 a^3 z^3+21 a z^3+25 z^3 a^{-1} +10 z^3 a^{-3} -5 a^4 z^2-17 a^2 z^2-6 z^2 a^{-2} -18 z^2-11 a^3 z-24 a z-18 z a^{-1} -5 z a^{-3} +7 a^4+15 a^2+3 a^{-2} +12+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                     0 5               1 1   0 3           1 1       0 1             1       1 -1       1 3 1         1 -3     1   1           2 -5     3 1             2 -7 1   1               2 -9 1                   1

Integral Khovanov Homology (db, data source)

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