L10a159

L10a158

L10a160

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Link Presentations

[edit Notes on L10a159's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,13,12 X2738 X6,13,1,14
Gauss code	{1, -9, 3, -4, 2, -10}, {9, -1, 7, -6, 8, -5}, {10, -3, 4, -2, 5, -7, 6, -8}

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(2)^2 t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+4 t(1) t(2) t(3)^2-t(2) t(3)^2-t(1)^2 t(3)+t(1) t(2)^2 t(3)-t(2)^2 t(3)+2 t(1) t(3)+t(1)^2 t(2) t(3)-4 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-13} -3 q^{-12} +6 q^{-11} -8 q^{-10} +11 q^{-9} -10 q^{-8} +10 q^{-7} -7 q^{-6} +5 q^{-5} -2 q^{-4} + q^{-3} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{12}+a^{12} z^{-2} +2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-2 a^{10} z^{-2} -9 a^{10}+2 z^6 a^8+9 z^4 a^8+13 z^2 a^8+a^8 z^{-2} +7 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^4-a^{16} z^2+3 a^{15} z^5-3 a^{15} z^3+5 a^{14} z^6-7 a^{14} z^4+5 a^{14} z^2-2 a^{14}+5 a^{13} z^7-6 a^{13} z^5+3 a^{13} z^3+3 a^{12} z^8+a^{12} z^6-8 a^{12} z^4+5 a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^9+6 a^{11} z^7-17 a^{11} z^5+17 a^{11} z^3-9 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-16 a^{10} z^6+22 a^{10} z^4-23 a^{10} z^2-2 a^{10} z^{-2} +11 a^{10}+a^9 z^9+3 a^9 z^7-14 a^9 z^5+15 a^9 z^3-9 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+18 a^8 z^4-18 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 3     3 -11               4 2     -2 -13             6 3       3 -15           4 4         0 -17         7 6           1 -19       3 6             3 -21     3 5               -2 -23   1 4                 3 -25   2                   -2 -27 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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