L11a502

L11a501

L11a503

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Link Presentations

[edit Notes on L11a502's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X20,15,21,16 X14,5,15,6 X4,13,5,14 X12,19,7,20 X18,11,19,12 X22,17,13,18 X16,21,17,22 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -6}, {5, -4, 3, -9, 8, -7, 6, -3, 9, -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)^2 t(3)^3+2 t(1) t(2)^2 t(3)^3-t(2)^2 t(3)^3+t(1) t(3)^3+2 t(1)^2 t(2) t(3)^3-3 t(1) t(2) t(3)^3+t(2) t(3)^3+t(1)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+4 t(1) t(2) t(3)^2-2 t(2) t(3)^2-t(1)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+2 t(1) t(3)+2 t(1)^2 t(2) t(3)-4 t(1) t(2) t(3)+2 t(2) t(3)+t(1)^2-t(1) t(2)^2+t(2)^2-2 t(1)-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} -2 q^{-4} +6 q^{-5} -9 q^{-6} +14 q^{-7} -15 q^{-8} +17 q^{-9} -14 q^{-10} +11 q^{-11} -7 q^{-12} +3 q^{-13} - q^{-14} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} -a^{12} z^4-a^{12} z^2+4 a^{12} z^{-2} +4 a^{12}+a^{10} z^6+a^{10} z^4-6 a^{10} z^2-5 a^{10} z^{-2} -10 a^{10}+2 a^8 z^6+7 a^8 z^4+7 a^8 z^2+2 a^8 z^{-2} +4 a^8+a^6 z^6+4 a^6 z^4+5 a^6 z^2+2 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{17}-2 z^3 a^{17}+3 z^6 a^{16}-5 z^4 a^{16}+6 z^7 a^{15}-14 z^5 a^{15}+12 z^3 a^{15}-5 z a^{15}+a^{15} z^{-1} +7 z^8 a^{14}-18 z^6 a^{14}+22 z^4 a^{14}-12 z^2 a^{14}-a^{14} z^{-2} +4 a^{14}+4 z^9 a^{13}-2 z^7 a^{13}-13 z^5 a^{13}+30 z^3 a^{13}-21 z a^{13}+5 a^{13} z^{-1} +z^{10} a^{12}+9 z^8 a^{12}-30 z^6 a^{12}+44 z^4 a^{12}-33 z^2 a^{12}-4 a^{12} z^{-2} +17 a^{12}+6 z^9 a^{11}-8 z^7 a^{11}-9 z^5 a^{11}+35 z^3 a^{11}-33 z a^{11}+9 a^{11} z^{-1} +z^{10} a^{10}+5 z^8 a^{10}-17 z^6 a^{10}+27 z^4 a^{10}-33 z^2 a^{10}-5 a^{10} z^{-2} +20 a^{10}+2 z^9 a^9+2 z^7 a^9-16 z^5 a^9+21 z^3 a^9-16 z a^9+5 a^9 z^{-1} +3 z^8 a^8-7 z^6 a^8+6 z^4 a^8-7 z^2 a^8-2 a^8 z^{-2} +6 a^8+2 z^7 a^7-5 z^5 a^7+2 z^3 a^7+z a^7+z^6 a^6-4 z^4 a^6+5 z^2 a^6-2 a^6 }[/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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     2 1 -1 -9                   4     4 -11                 5 2     -3 -13               9 4       5 -15             7 6         -1 -17           10 8           2 -19         6 9             3 -21       5 8               -3 -23     2 6                 4 -25   1 5                   -4 -27   2                     2 -29 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

L11a501 L11a503