L11n152

L11n151

L11n153

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Link Presentations

[edit Notes on L11n152's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X19,7,20,22 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 X2,18,3,17
Gauss code	{1, -11, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, -9, 10, 7, -8, 11, -2, -4, 9, -10, 4}

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

Integral Khovanov Homology (db, data source)

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

L11n151 L11n153