L11a354

L11a353

L11a355

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Link Presentations

[edit Notes on L11a354's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X8394 X16,11,17,12 X14,8,15,7 X6,16,7,15 X22,17,11,18 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {4, -1, 2, -5, 6, -4, 7, -9, 8, -10, 11, -7}

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)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+3 q^{3/2}-8 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-2 z a^7+2 z^5 a^5+5 z^3 a^5+3 z a^5-z^7 a^3-3 z^5 a^3-3 z^3 a^3-2 z a^3+2 z^5 a+5 z^3 a+3 z a+a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+2 z^3 a^9-6 z^6 a^8+5 z^4 a^8-2 z^2 a^8-10 z^7 a^7+17 z^5 a^7-16 z^3 a^7+4 z a^7-11 z^8 a^6+21 z^6 a^6-17 z^4 a^6+5 z^2 a^6-7 z^9 a^5+4 z^7 a^5+20 z^5 a^5-23 z^3 a^5+6 z a^5-2 z^{10} a^4-13 z^8 a^4+46 z^6 a^4-39 z^4 a^4+12 z^2 a^4-11 z^9 a^3+24 z^7 a^3-2 z^5 a^3-11 z^3 a^3+4 z a^3-2 z^{10} a^2-5 z^8 a^2+29 z^6 a^2-26 z^4 a^2+6 z^2 a^2-4 z^9 a+9 z^7 a+2 z^5 a-12 z^3 a+6 z a-a z^{-1} -3 z^8+10 z^6-10 z^4+2 z^2+1-z^7 a^{-1} +4 z^5 a^{-1} -6 z^3 a^{-1} +4 z a^{-1} - a^{-1} z^{-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                   6 1   5 0                 6 2     -4 -2               11 6       5 -4             10 8         -2 -6           9 9           0 -8         8 10             2 -10       5 9               -4 -12     2 8                 6 -14   1 5                   -4 -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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/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}^{2}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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