L11a76

L11a75

L11a77

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Link Presentations

[edit Notes on L11a76's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,15,21,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X22,13,5,14 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, 8, -9, 3, -4, 7, -6, 5, -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{4 t(2)^3+6 t(1) t(2)^2-9 t(2)^2-9 t(1) t(2)+6 t(2)+4 t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{9}{q^{17/2}}-\frac{7}{q^{19/2}}+\frac{5}{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}+a^{11} z^{-1} -2 z^3 a^9+z a^9+2 a^9 z^{-1} -4 z^3 a^7-4 z a^7-2 a^7 z^{-1} -3 z^3 a^5-2 z a^5-z^3 a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^8-6 a^{14} z^6+13 a^{14} z^4-12 a^{14} z^2+4 a^{14}+2 a^{13} z^9-10 a^{13} z^7+15 a^{13} z^5-6 a^{13} z^3-a^{13} z^{-1} +a^{12} z^{10}+a^{12} z^8-23 a^{12} z^6+48 a^{12} z^4-34 a^{12} z^2+9 a^{12}+6 a^{11} z^9-23 a^{11} z^7+20 a^{11} z^5+3 a^{11} z^3-3 a^{11} z-a^{11} z^{-1} +a^{10} z^{10}+7 a^{10} z^8-38 a^{10} z^6+46 a^{10} z^4-19 a^{10} z^2+4 a^{10}+4 a^9 z^9-4 a^9 z^7-19 a^9 z^5+26 a^9 z^3-12 a^9 z+2 a^9 z^{-1} +7 a^8 z^8-13 a^8 z^6-2 a^8 z^4+5 a^8 z^2-2 a^8+9 a^7 z^7-18 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +8 a^6 z^6-10 a^6 z^4+2 a^6 z^2+6 a^5 z^5-5 a^5 z^3+2 a^5 z+3 a^4 z^4+a^3 z^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                     3 1 -2 -6                   3     3 -8                 5 3     -2 -10               6 3       3 -12             6 6         0 -14           6 5           1 -16         3 6             3 -18       4 6               -2 -20     1 3                 2 -22   1 4                   -3 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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