L11n380

L11n379

L11n381

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Link Presentations

[edit Notes on L11n380's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, -6, 9}

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) (w-1)^2 (v+w)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +2 q^{-9} -4 q^{-8} +5 q^{-7} -4 q^{-6} +6 q^{-5} -4 q^{-4} +4 q^{-3} - q^{-2} + q^{-1} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+3 z^2 a^8+4 a^8 z^{-2} +6 a^8-2 z^4 a^6-7 z^2 a^6-5 a^6 z^{-2} -10 a^6+3 z^2 a^4+2 a^4 z^{-2} +4 a^4+z^2 a^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{11}-5 z^5 a^{11}+8 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +2 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+z^9 a^9-17 z^5 a^9+31 z^3 a^9-20 z a^9+5 a^9 z^{-1} +5 z^8 a^8-23 z^6 a^8+31 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+z^9 a^7+z^7 a^7-21 z^5 a^7+39 z^3 a^7-30 z a^7+9 a^7 z^{-1} +3 z^8 a^6-14 z^6 a^6+23 z^4 a^6-23 z^2 a^6-5 a^6 z^{-2} +16 a^6+2 z^7 a^5-9 z^5 a^5+17 z^3 a^5-15 z a^5+5 a^5 z^{-1} +3 z^4 a^4-7 z^2 a^4-2 a^4 z^{-2} +6 a^4+z^3 a^3+z^2 a^2-a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   1 1 -3                 2 2 0 -5               2   1 3 -7             3 3     0 -9           4 2       2 -11         2 5 1       2 -13       3 3 1         1 -15     1 2             1 -17   1 3               -2 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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