L11n444

L11n443

L11n445

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Link Presentations

[edit Notes on L11n444's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X13,20,14,21 X19,15,20,22 X21,12,22,13 X17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -3, 10, -8, 11}, {7, -6, -11, 3, -9, 8, -10, 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(1) t(4)^2 t(3)^2-t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(2) t(3)^2+t(1) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)+t(2) t(4)^2 t(3)-t(4)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-2 t(1) t(4) t(3)-2 t(2) t(4) t(3)-t(1) t(4)^2-t(1)+t(1) t(2)-t(2)+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} +a^9 z^{-3} -4 z a^7-7 a^7 z^{-1} -3 a^7 z^{-3} +5 z^3 a^5+14 z a^5+12 a^5 z^{-1} +3 a^5 z^{-3} -2 z^5 a^3-8 z^3 a^3-13 z a^3-7 a^3 z^{-1} -a^3 z^{-3} +2 z^3 a+3 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-10 z^3 a^9+10 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -2 z^8 a^8+7 z^6 a^8-4 z^4 a^8-7 z^2 a^8-3 a^8 z^{-2} +9 a^8-z^9 a^7-4 z^7 a^7+31 z^5 a^7-50 z^3 a^7+35 z a^7-14 a^7 z^{-1} +3 a^7 z^{-3} -7 z^8 a^6+21 z^6 a^6-4 z^4 a^6-24 z^2 a^6-6 a^6 z^{-2} +21 a^6-z^9 a^5-11 z^7 a^5+54 z^5 a^5-74 z^3 a^5+49 z a^5-18 a^5 z^{-1} +3 a^5 z^{-3} -5 z^8 a^4+9 z^6 a^4+11 z^4 a^4-28 z^2 a^4-3 a^4 z^{-2} +18 a^4-8 z^7 a^3+27 z^5 a^3-39 z^3 a^3+30 z a^3-11 a^3 z^{-1} +a^3 z^{-3} -5 z^6 a^2+11 z^4 a^2-14 z^2 a^2+6 a^2-z^5 a-5 z^3 a+6 z a-2 a z^{-1} -3 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   2 -2 0                 3   3 -2               4 3   -1 -4             6 3 1   4 -6           5 7       2 -8         5 4 1       2 -10       4 8           4 -12     2 2             0 -14   1 5               4 -16   1                 -1 -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{ 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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/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} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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]

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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