L11n449

L11n448

L11n450

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Link Presentations

[edit Notes on L11n449's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X22,17,19,18 X18,21,9,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, 9, -8}, {11, -2, 4, 5, -3, -7, 6, -4, 8, -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{(t(4)-1) \left(-t(1) t(4)^2+t(1) t(2) t(4)^2-2 t(2) t(4)^2+t(4)^2+t(1) t(4)-t(2) t(3) t(4)+2 t(1) t(3)-t(1) t(2) t(3)+t(2) t(3)-t(3)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -3 a^7 z-5 a^7 z^{-1} +3 a^5 z^3+3 a^5 z^{-3} +6 a^5 z+6 a^5 z^{-1} -a^3 z^5-2 a^3 z^3-a^3 z^{-3} -2 a^3 z-a^3 z^{-1} +a z^3-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-6 a^9 z^5+13 a^9 z^3-a^9 z^{-3} -13 a^9 z+6 a^9 z^{-1} +a^8 z^8-2 a^8 z^6-6 a^8 z^4+17 a^8 z^2+3 a^8 z^{-2} -13 a^8+a^7 z^9-14 a^7 z^5+29 a^7 z^3-3 a^7 z^{-3} -27 a^7 z+14 a^7 z^{-1} +5 a^6 z^8-14 a^6 z^6-3 a^6 z^4+28 a^6 z^2+6 a^6 z^{-2} -24 a^6+a^5 z^9+5 a^5 z^7-26 a^5 z^5+30 a^5 z^3-3 a^5 z^{-3} -22 a^5 z+12 a^5 z^{-1} +4 a^4 z^8-9 a^4 z^6+11 a^4 z^2+3 a^4 z^{-2} -11 a^4+6 a^3 z^7-18 a^3 z^5+19 a^3 z^3-a^3 z^{-3} -8 a^3 z+3 a^3 z^{-1} +3 a^2 z^6-3 a^2 z^4+a^2 z^2+a^2+5 a z^3-a z^{-1} +z^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   1 -1 0                 4   4 -2               5 4   -1 -4             3 2 1   2 -6           4 5       1 -8         5 3         2 -10       4 8           4 -12     1 1             0 -14     4               4 -16 1 1                 0 -18 1                   1

Integral Khovanov Homology (db, data source)

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