L11n389

L11n388

L11n390

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Link Presentations

[edit Notes on L11n389's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -7, 8, -10, 4, -6, 9, -11, -2, 3, 10}

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{(t(3)-1) \left(t(1) t(3)^4-t(1) t(3)^3+2 t(1) t(2) t(3)^3+t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2-t(3)^2-t(2) t(3)+2 t(3)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -7 q^{-8} +8 q^{-9} -6 q^{-10} +3 q^{-11} - q^{-12} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +z^2 a^{12}+4 a^{12} z^{-2} +6 a^{12}-3 z^4 a^{10}-13 z^2 a^{10}-5 a^{10} z^{-2} -15 a^{10}+2 z^6 a^8+9 z^4 a^8+12 z^2 a^8+2 a^8 z^{-2} +8 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6+a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{15}-2 z a^{15}+a^{15} z^{-1} +3 z^4 a^{14}-3 z^2 a^{14}-a^{14} z^{-2} +2 a^{14}+2 z^7 a^{13}-7 z^5 a^{13}+18 z^3 a^{13}-15 z a^{13}+5 a^{13} z^{-1} +3 z^8 a^{12}-12 z^6 a^{12}+26 z^4 a^{12}-25 z^2 a^{12}-4 a^{12} z^{-2} +14 a^{12}+z^9 a^{11}+3 z^7 a^{11}-22 z^5 a^{11}+44 z^3 a^{11}-33 z a^{11}+9 a^{11} z^{-1} +6 z^8 a^{10}-24 z^6 a^{10}+42 z^4 a^{10}-42 z^2 a^{10}-5 a^{10} z^{-2} +21 a^{10}+z^9 a^9+3 z^7 a^9-21 z^5 a^9+30 z^3 a^9-20 z a^9+5 a^9 z^{-1} +3 z^8 a^8-11 z^6 a^8+15 z^4 a^8-16 z^2 a^8-2 a^8 z^{-2} +9 a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6 }[/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 -5                   1 1 -7                 2 1 -1 -9               3     3 -11             3 2     -1 -13           6 3       3 -15         3 5         2 -17       5 4           1 -19     1 3             2 -21   2 5               -3 -23   2                 2 -25 1                   -1

Integral Khovanov Homology (db, data source)

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