L11n348

L11n347

L11n349

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Link Presentations

[edit Notes on L11n348's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 9, -6, 7, -3, 8}, {11, -2, -7, 6, -5, 4, -8, 3, -9, 5}

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 v^2 w^3-2 u v^2 w^2-u v w^3+u v w+u w^2-u w+v^3 w^2-v^3 w-v^2 w^2+v^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} + q^{-6} +2 q^{-7} -2 q^{-8} +2 q^{-9} - q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+2 z^2 a^8+a^8 z^{-2} +2 a^8-2 a^6 z^{-2} -3 a^6-z^6 a^4-4 z^4 a^4-2 z^2 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+16 a^{10} z^4-10 a^{10} z^2+3 a^{10}+a^9 z^9-4 a^9 z^7-3 a^9 z^5+14 a^9 z^3-7 a^9 z+3 a^8 z^8-21 a^8 z^6+40 a^8 z^4-29 a^8 z^2-a^8 z^{-2} +11 a^8+a^7 z^9-5 a^7 z^7-a^7 z^5+16 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+27 a^6 z^4-24 a^6 z^2-2 a^6 z^{-2} +11 a^6+2 a^5 z^7-11 a^5 z^5+14 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +a^4 z^8-3 a^4 z^6-a^4 z^4-a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^7-8 a^3 z^5+6 a^3 z^3-a^3 z+a^2 z^6-4 a^2 z^4+4 a^2 z^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 1 2 χ 1                       1 1 -1                     1   -1 -3                   2 1   1 -5               1 2 2     1 -7             1 2 1       0 -9           2 2 2         2 -11         2 5 2           1 -13       1 1 3             3 -15     1 2 1               0 -17   1 1                   0 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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