L11n370

L11n369

L11n371

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Link Presentations

[edit Notes on L11n370's Link Presentations]

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

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-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-6} +5 q^{-5} -7 q^{-4} +q^3+11 q^{-3} -3 q^2-10 q^{-2} +6 q+11 q^{-1} -9 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} -a^4 z^4-a^4 z^2-2 a^4 z^{-2} -2 a^4+a^2 z^6+3 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2+ a^{-2} -2 z^4-4 z^2-2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^3+5 a^6 z^4-a^6 z^2+a^6 z^{-2} -2 a^6+2 a^5 z^7+2 a^5 z^3+2 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-4 a^4 z^6+6 a^4 z^4-3 a^4 z^2+2 a^4 z^{-2} -2 a^4+a^3 z^9+6 a^3 z^7-16 a^3 z^5+12 a^3 z^3-2 a^3 z-2 a^3 z^{-1} +6 a^2 z^8-9 a^2 z^6+z^6 a^{-2} -3 a^2 z^4-3 z^4 a^{-2} +3 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} -a^2- a^{-2} +a z^9+7 a z^7+3 z^7 a^{-1} -25 a z^5-9 z^5 a^{-1} +19 a z^3+8 z^3 a^{-1} -6 a z-2 z a^{-1} +3 z^8-4 z^6-7 z^4+8 z^2-1 }[/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               4 1   3 1             5 2     -3 -1           6 4       2 -3         6 7         1 -5       5 4           1 -7     2 6             4 -9   3 5               -2 -11   4                 4 -13 1                   -1

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