L11a529

L11a528

L11a530

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Link Presentations

[edit Notes on L11a529's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,13,7,14 X16,7,17,8 X22,17,13,18 X18,5,19,6 X20,9,21,10 X10,19,11,20 X4,11,5,12 X2,15,3,16 X6,21,1,22
Gauss code	{1, -10, 2, -9, 6, -11}, {4, -1, 7, -8, 9, -3}, {3, -2, 10, -4, 5, -6, 8, -7, 11, -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)^2 t(3)^3+t(2)^2 t(3)^3-2 t(1) t(3)^3+2 t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3-3 t(1)^2 t(3)^2+3 t(1) t(2)^2 t(3)^2-3 t(2)^2 t(3)^2+5 t(1) t(3)^2+3 t(1)^2 t(2) t(3)^2-7 t(1) t(2) t(3)^2+5 t(2) t(3)^2-2 t(3)^2+3 t(1)^2 t(3)+2 t(1)^2 t(2)^2 t(3)-5 t(1) t(2)^2 t(3)+3 t(2)^2 t(3)-3 t(1) t(3)-5 t(1)^2 t(2) t(3)+7 t(1) t(2) t(3)-3 t(2) t(3)-t(1)^2-t(1)^2 t(2)^2+2 t(1) t(2)^2-t(2)^2+2 t(1)^2 t(2)-2 t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-13} +5 q^{-12} -10 q^{-11} +17 q^{-10} -22 q^{-9} +26 q^{-8} -25 q^{-7} +23 q^{-6} -16 q^{-5} +10 q^{-4} -4 q^{-3} + q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} \left(-z^2\right)+a^{12} z^{-2} +2 a^{12}+a^{10} z^4-6 a^{10} z^2-2 a^{10} z^{-2} -9 a^{10}+6 a^8 z^4+11 a^8 z^2+a^8 z^{-2} +7 a^8+4 a^6 z^4+4 a^6 z^2+a^4 z^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{15}-2 z^5 a^{15}+z^3 a^{15}+5 z^8 a^{14}-15 z^6 a^{14}+13 z^4 a^{14}-z^2 a^{14}-2 a^{14}+8 z^9 a^{13}-23 z^7 a^{13}+17 z^5 a^{13}-z^3 a^{13}+4 z^{10} a^{12}+6 z^8 a^{12}-46 z^6 a^{12}+46 z^4 a^{12}-11 z^2 a^{12}-a^{12} z^{-2} +3 a^{12}+21 z^9 a^{11}-53 z^7 a^{11}+26 z^5 a^{11}+9 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} +4 z^{10} a^{10}+20 z^8 a^{10}-76 z^6 a^{10}+70 z^4 a^{10}-33 z^2 a^{10}-2 a^{10} z^{-2} +11 a^{10}+13 z^9 a^9-13 z^7 a^9-17 z^5 a^9+19 z^3 a^9-9 z a^9+2 a^9 z^{-1} +19 z^8 a^8-35 z^6 a^8+28 z^4 a^8-19 z^2 a^8-a^8 z^{-2} +7 a^8+16 z^7 a^7-20 z^5 a^7+8 z^3 a^7+10 z^6 a^6-8 z^4 a^6+4 z^2 a^6+4 z^5 a^5+z^4 a^4 }[/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 -3                       1 1 -5                     4 1 -3 -7                   6     6 -9                 10 4     -6 -11               13 6       7 -13             12 10         -2 -15           14 13           1 -17         10 14             4 -19       7 12               -5 -21     4 11                 7 -23   1 6                   -5 -25   4                     4 -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=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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