L11n103

L11n102

L11n104

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Link Presentations

[edit Notes on L11n103's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, -5, 6, -8, 3, -4, 5, -9, 7, -6, 4}

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{3 t(1) t(2)^3+t(2)^3-3 t(1) t(2)^2-3 t(2)+t(1)+3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}+\frac{1}{q^{19/2}}-\frac{3}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{4}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +2 a^{11} z+a^{11} z^{-1} +a^9 z^3+4 a^9 z+2 a^9 z^{-1} -2 a^7 z^5-8 a^7 z^3-7 a^7 z-2 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^8-7 a^{14} z^6+15 a^{14} z^4-13 a^{14} z^2+4 a^{14}+a^{13} z^9-7 a^{13} z^7+13 a^{13} z^5-7 a^{13} z^3+a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-17 a^{12} z^6+41 a^{12} z^4-34 a^{12} z^2+9 a^{12}+a^{11} z^9-7 a^{11} z^7+12 a^{11} z^5-2 a^{11} z^3-a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+27 a^{10} z^4-17 a^{10} z^2+4 a^{10}+3 a^9 z^7-13 a^9 z^5+21 a^9 z^3-12 a^9 z+2 a^9 z^{-1} +a^8 z^8-a^8 z^6-4 a^8 z^4+5 a^8 z^2-2 a^8+3 a^7 z^7-11 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-5 a^6 z^4+a^6 z^2+a^5 z^5-3 a^5 z^3+a^5 z }[/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 -4                       1 1 -6                     2 1 -1 -8                   2     2 -10                 2 2     0 -12             1 4 2       1 -14             1 3         2 -16         1 4 3           0 -18       1   1             2 -20       1 3               -2 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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