L11a253

L11a252

L11a254

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Link Presentations

[edit Notes on L11a253's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,15,21,16 X16,8,17,7 X4,13,5,14 X6,20,7,19
Gauss code	{1, -2, 3, -10, 4, -11, 9, -7}, {5, -1, 2, -3, 10, -4, 8, -9, 6, -5, 11, -8, 7, -6}

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)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+3 t(1)^2 t(2)^4-t(1) t(2)^4+2 t(1)^3 t(2)^3-5 t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{10}{q^{9/2}}-\frac{12}{q^{7/2}}-q^{5/2}+\frac{11}{q^{5/2}}+3 q^{3/2}-\frac{10}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-5 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^9-a^5 z^7+7 a^3 z^7-a z^7-5 a^5 z^5+17 a^3 z^5-5 a z^5-7 a^5 z^3+16 a^3 z^3-7 a z^3-2 a^5 z+3 a^3 z-3 a z+a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+4 z^4 a^8-4 z^7 a^7+3 z^5 a^7+2 z^3 a^7-2 z a^7-4 z^8 a^6+6 z^6 a^6-3 z^4 a^6-4 z^9 a^5+12 z^7 a^5-17 z^5 a^5+11 z^3 a^5-z a^5-a^5 z^{-1} -2 z^{10} a^4+3 z^8 a^4+5 z^6 a^4-10 z^4 a^4+3 z^2 a^4+a^4-8 z^9 a^3+35 z^7 a^3-51 z^5 a^3+30 z^3 a^3-4 z a^3-a^3 z^{-1} -2 z^{10} a^2+4 z^8 a^2+8 z^6 a^2-16 z^4 a^2+6 z^2 a^2-4 z^9 a+18 z^7 a-24 z^5 a+14 z^3 a-4 z a-3 z^8+13 z^6-14 z^4+4 z^2-z^7 a^{-1} +4 z^5 a^{-1} -3 z^3 a^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   3 1   2 0                 4 2     -2 -2               6 3       3 -4             6 5         -1 -6           6 5           1 -8         4 6             2 -10       4 6               -2 -12     2 5                 3 -14   1 3                   -2 -16   2                     2 -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{ r=-7 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/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}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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}_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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