L11n371

L11n370

L11n372

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Link Presentations

[edit Notes on L11n371's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,22,16,17 X17,16,18,5 X9,19,10,18 X13,21,14,20 X19,13,20,12 X21,9,22,8 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {-5, 6, -8, 7, -9, 4}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 3, -4, 5}

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(3)-1) (t(1) t(3)+1) (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+7 q-9+12 q^{-1} -12 q^{-2} +12 q^{-3} -8 q^{-4} +6 q^{-5} -2 q^{-6} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^2-2 a^6+a^4 z^6+5 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +8 a^4-a^2 z^8-6 a^2 z^6-14 a^2 z^4-17 a^2 z^2-2 a^2 z^{-2} -11 a^2+z^6+4 z^4+6 z^2+ z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^7 z^3-2 a^7 z+a^6 z^6+6 a^6 z^4-8 a^6 z^2+2 a^6+5 a^5 z^7-11 a^5 z^5+17 a^5 z^3-7 a^5 z+6 a^4 z^8-18 a^4 z^6+33 a^4 z^4-28 a^4 z^2-a^4 z^{-2} +10 a^4+2 a^3 z^9+5 a^3 z^7-24 a^3 z^5+30 a^3 z^3-14 a^3 z+2 a^3 z^{-1} +10 a^2 z^8-29 a^2 z^6+z^6 a^{-2} +33 a^2 z^4-3 z^4 a^{-2} -26 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +12 a^2- a^{-2} +2 a z^9+3 a z^7+3 z^7 a^{-1} -21 a z^5-8 z^5 a^{-1} +21 a z^3+5 z^3 a^{-1} -10 a z-z a^{-1} +2 a z^{-1} +4 z^8-9 z^6+3 z^4-3 z^2- z^{-2} +4 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               5 1   4 1             5 3     -2 -1           7 4       3 -3         6 6         0 -5       6 6           0 -7     3 7             4 -9   3 5               -2 -11   4                 4 -13 2                   -2

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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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