L11n427

L11n426

L11n428

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Link Presentations

[edit Notes on L11n427's Link Presentations]

Planar diagram presentation	X8192 X11,18,12,19 X3,10,4,11 X19,2,20,3 X7,16,8,17 X20,9,21,10 X17,12,18,7 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22
Gauss code	{1, 4, -3, -10, 9, -11}, {-5, -1, 6, 3, -2, 7}, {10, -9, 8, 5, -7, 2, -4, -6, 11, -8}

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)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(2)^2 t(3)^2-t(1) t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+4 t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-4 t(1) t(2) t(3)+2 t(2) t(3)-t(3)+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-7} -3 q^{-6} +7 q^{-5} -8 q^{-4} +10 q^{-3} -q^2-9 q^{-2} +3 q+8 q^{-1} -5 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +a^8-a^6 z^4-4 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^6+4 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +4 a^4+a^2 z^6+3 a^2 z^4+2 a^2 z^2-z^4-2 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^8 z^4-8 a^8 z^2-a^8 z^{-2} +6 a^8+a^7 z^7-a^7 z^5+3 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-7 a^6 z^6+20 a^6 z^4-25 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^9-a^5 z^5+7 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-15 a^4 z^6+27 a^4 z^4-22 a^4 z^2-a^4 z^{-2} +8 a^4+a^3 z^9+3 a^3 z^7-10 a^3 z^5+11 a^3 z^3-3 a^3 z+3 a^2 z^8-5 a^2 z^6+3 a^2 z^4-2 a^2 z^2+a^2+4 a z^7-9 a z^5+z^5 a^{-1} +5 a z^3-2 z^3 a^{-1} -a z+3 z^6-7 z^4+3 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 χ 5                   1 -1 3                 2   2 1               3 1   -2 -1             5 2     3 -3           5 4       -1 -5         5 4         1 -7       4 6           2 -9     3 4             -1 -11   1 5               4 -13 1 2                 -1 -15 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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