L11a471

L11a470

L11a472

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Link Presentations

[edit Notes on L11a471's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X20,9,21,10 X10,19,11,20 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, 7, -6, 10, -9}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -8, 9, -10}

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(3)-1)^2 (t(2)+t(3)-1) (t(3) t(2)-t(2)-t(3))}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5- q^{-5} +9 q^4+4 q^{-4} -14 q^3-9 q^{-3} +21 q^2+15 q^{-2} -22 q-20 q^{-1} +24 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} -a^4 z^2+z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -z^6 a^{-2} +2 a^2 z^4-2 z^4 a^{-2} +a^2 z^2-4 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2-4 a^{-2} -z^6+3 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -2 z^4 a^{-6} +4 z^7 a^{-5} +a^5 z^5-9 z^5 a^{-5} -a^5 z^3+4 z^3 a^{-5} +8 z^8 a^{-4} +4 a^4 z^6-22 z^6 a^{-4} -5 a^4 z^4+21 z^4 a^{-4} +2 a^4 z^2-9 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +7 z^9 a^{-3} +8 a^3 z^7-12 z^7 a^{-3} -12 a^3 z^5+2 z^5 a^{-3} +7 a^3 z^3+4 z^3 a^{-3} -a^3 z-2 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +9 a^2 z^8+15 z^8 a^{-2} -10 a^2 z^6-50 z^6 a^{-2} +3 a^2 z^4+54 z^4 a^{-2} -3 a^2 z^2-27 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2+7 a^{-2} +6 a z^9+13 z^9 a^{-1} +4 a z^7-20 z^7 a^{-1} -21 a z^5+3 z^5 a^{-1} +14 a z^3+6 z^3 a^{-1} -3 a z-4 z a^{-1} +2 a^{-1} z^{-1} +2 z^{10}+16 z^8-41 z^6+39 z^4-23 z^2- z^{-2} +7 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 8 3     -5 5               13 6       7 3             11 10         -1 1           13 11           2 -1         9 13             4 -3       6 11               -5 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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