L11n267

L11n266

L11n268

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Link Presentations

[edit Notes on L11n267's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -6, 8, -7, 5, -9, 6, -8, 7}

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{u v w^3-u v w^2+u v+u w^5-u w^4+u w-u+v w^5-v w^4+v w-v-w^5+w^3-w^2}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+q^4-q^3-q^2+q-1+4 q^{-1} -2 q^{-2} +4 q^{-3} - q^{-4} + q^{-5} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z^6-3 a^2 z^4-z^4 a^{-2} +12 z^4+a^4 z^2-14 a^2 z^2-6 z^2 a^{-2} -z^2 a^{-4} +23 z^2+4 a^4-18 a^2-7 a^{-2} +21+3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} +7 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7-a z^7+3 z^7 a^{-1} -7 a^4 z^6-31 a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -26 z^6-2 a^3 z^5-21 a z^5-20 z^5 a^{-1} +z^5 a^{-5} +18 a^4 z^4+61 a^2 z^4+3 z^4 a^{-2} -4 z^4 a^{-4} +50 z^4+21 a^3 z^3+53 a z^3+33 z^3 a^{-1} -3 z^3 a^{-3} -4 z^3 a^{-5} -22 a^4 z^2-54 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -42 z^2-24 a^3 z-45 a z-21 z a^{-1} +3 z a^{-3} +3 z a^{-5} +13 a^4+28 a^2+7 a^{-2} + a^{-4} +22+8 a^3 z^{-1} +15 a z^{-1} +7 a^{-1} z^{-1} - a^{-3} z^{-1} - a^{-5} z^{-1} -3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} -7 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                         0 7                   1 1   0 5               3 1       -2 3             1 2 1       0 1           3 4 1         0 -1         1 1 3           3 -3       1 3               2 -5     3 1                 2 -7   1 4                   3 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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