L11a331

L11a330

L11a332

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Link Presentations

[edit Notes on L11a331's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X22,18,9,17 X18,14,19,13 X14,22,15,21 X16,7,17,8 X8,9,1,10 X6,15,7,16 X4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -8, 5, -6, 11, -4, 7, -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{t(1)^3 t(2)^5-3 t(1)^3 t(2)^4+3 t(1)^2 t(2)^4+3 t(1)^3 t(2)^3-6 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-6 t(1) t(2)^2+3 t(2)^2+3 t(1) t(2)-3 t(2)+1}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^7+5 a^7 z^5+8 a^7 z^3+6 a^7 z+2 a^7 z^{-1} -a^5 z^9-7 a^5 z^7-18 a^5 z^5-22 a^5 z^3-13 a^5 z-3 a^5 z^{-1} +a^3 z^7+5 a^3 z^5+7 a^3 z^3+3 a^3 z+a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{13}-3 z^4 a^{12}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}-8 z^6 a^{10}+10 z^4 a^{10}-2 z^2 a^{10}-9 z^7 a^9+17 z^5 a^9-7 z^3 a^9+3 z a^9-8 z^8 a^8+18 z^6 a^8-6 z^4 a^8+z^2 a^8-6 z^9 a^7+17 z^7 a^7-12 z^5 a^7+10 z^3 a^7-8 z a^7+2 a^7 z^{-1} -2 z^{10} a^6-2 z^8 a^6+31 z^6 a^6-43 z^4 a^6+18 z^2 a^6-3 a^6-9 z^9 a^5+42 z^7 a^5-62 z^5 a^5+39 z^3 a^5-16 z a^5+3 a^5 z^{-1} -2 z^{10} a^4+5 z^8 a^4+10 z^6 a^4-32 z^4 a^4+20 z^2 a^4-3 a^4-3 z^9 a^3+16 z^7 a^3-27 z^5 a^3+16 z^3 a^3-3 z a^3+a^3 z^{-1} -z^8 a^2+5 z^6 a^2-8 z^4 a^2+5 z^2 a^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   3 1   -2 -4                 5 2     3 -6               6 4       -2 -8             6 4         2 -10           6 6           0 -12         5 6             -1 -14       3 6               3 -16     3 5                 -2 -18     3                   3 -20 1 3                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

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