L11n261

L11n260

L11n262

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Link Presentations

[edit Notes on L11n261's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,20,9,19 X20,16,21,15 X16,22,17,21 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 7, -8, 9, -5, 6, -7, 8, -6}

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+u v-u w^3+2 u w^2-2 u w+u-v w^3+2 v w^2-2 v w+v-w^3+w^2-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{10}+q^9-q^8-q^7+3 q^6-3 q^5+5 q^4-3 q^3+5 q^2-2 q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-10} z^{-2} - a^{-10} +2 z^2 a^{-8} +3 a^{-8} z^{-2} +5 a^{-8} -2 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} -z^6 a^{-4} -4 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-2} +z^6 a^{-4} -2 z^6 a^{-6} -8 z^6 a^{-8} -6 z^6 a^{-10} -6 z^5 a^{-3} -15 z^5 a^{-5} -20 z^5 a^{-7} -17 z^5 a^{-9} -6 z^5 a^{-11} -4 z^4 a^{-2} -14 z^4 a^{-4} -3 z^4 a^{-6} +15 z^4 a^{-8} +8 z^4 a^{-10} +z^3 a^{-3} +16 z^3 a^{-5} +42 z^3 a^{-7} +37 z^3 a^{-9} +10 z^3 a^{-11} +6 z^2 a^{-2} +12 z^2 a^{-4} +2 z^2 a^{-6} -7 z^2 a^{-8} -3 z^2 a^{-10} +4 z a^{-3} -10 z a^{-5} -34 z a^{-7} -27 z a^{-9} -7 z a^{-11} -4 a^{-2} -3 a^{-4} +4 a^{-6} +5 a^{-8} + a^{-10} -2 a^{-3} z^{-1} +2 a^{-5} z^{-1} +10 a^{-7} z^{-1} +8 a^{-9} z^{-1} +2 a^{-11} z^{-1} + a^{-2} z^{-2} + a^{-4} z^{-2} -2 a^{-6} z^{-2} -3 a^{-8} z^{-2} - a^{-10} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 21                       1 -1 19                         0 17                   1 1   0 15               3 1       -2 13             1   1       2 11           4 5 1         0 9         2 1 1           2 7       1 4 1             2 5     4 2                 2 3   1 4                   3 1   1                     -1 -1 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=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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