L11a37

L11a36

L11a38

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Link Presentations

[edit Notes on L11a37's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,15,19,16 X16,7,17,8 X8,17,9,18 X22,11,5,12 X20,13,21,14 X14,19,15,20 X12,21,13,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 7, -8, 3, -4, 5, -3, 8, -7, 9, -6}

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(2)^3+4 t(1) t(2)^2-6 t(2)^2-6 t(1) t(2)+4 t(2)+3 t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +3 z a^{11}+2 a^{11} z^{-1} -2 z^3 a^9-2 z a^9-a^9 z^{-1} -2 z^3 a^7+a^7 z^{-1} -2 z^3 a^5-2 z a^5-a^5 z^{-1} -z^3 a^3-z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{14}+6 z^6 a^{14}-12 z^4 a^{14}+10 z^2 a^{14}-3 a^{14}-2 z^9 a^{13}+11 z^7 a^{13}-18 z^5 a^{13}+10 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^{10} a^{12}+z^8 a^{12}+16 z^6 a^{12}-39 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-5 z^9 a^{11}+25 z^7 a^{11}-38 z^5 a^{11}+23 z^3 a^{11}-8 z a^{11}+2 a^{11} z^{-1} -z^{10} a^{10}-z^8 a^{10}+19 z^6 a^{10}-31 z^4 a^{10}+18 z^2 a^{10}-4 a^{10}-3 z^9 a^9+11 z^7 a^9-14 z^5 a^9+13 z^3 a^9-5 z a^9+a^9 z^{-1} -3 z^8 a^8+6 z^6 a^8-3 z^7 a^7+3 z^5 a^7+4 z^3 a^7-4 z a^7+a^7 z^{-1} -3 z^6 a^6+2 z^4 a^6+z^2 a^6-a^6-3 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3 }[/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 -2                       1 1 -4                     2 1 -1 -6                   2     2 -8                 3 2     -1 -10               4 2       2 -12             4 4         0 -14           4 3           1 -16         2 4             2 -18       3 4               -1 -20     1 2                 1 -22   1 3                   -2 -24   1                     1 -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=-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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