L11a75

L11a74

L11a76

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Link Presentations

[edit Notes on L11a75's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,15,21,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X22,13,5,14 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 11, -2, 8, -9, 3, -4, 6, -5, 7, -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{2 t(2)^5+3 t(1) t(2)^4-4 t(2)^4-4 t(1) t(2)^3+4 t(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+3 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{5}{q^{23/2}}-\frac{7}{q^{21/2}}+\frac{9}{q^{19/2}}-\frac{11}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{6}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-2 a^{13} z^{-1} +3 z^3 a^{11}+8 z a^{11}+4 a^{11} z^{-1} -2 z^5 a^9-6 z^3 a^9-4 z a^9-a^9 z^{-1} -2 z^5 a^7-6 z^3 a^7-4 z a^7-a^7 z^{-1} -z^5 a^5-3 z^3 a^5-z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{16}+4 z^4 a^{16}-5 z^2 a^{16}+2 a^{16}-2 z^7 a^{15}+6 z^5 a^{15}-3 z^3 a^{15}-z a^{15}-2 z^8 a^{14}+3 z^6 a^{14}+5 z^4 a^{14}-5 z^2 a^{14}+a^{14}-2 z^9 a^{13}+5 z^7 a^{13}-7 z^5 a^{13}+13 z^3 a^{13}-8 z a^{13}+2 a^{13} z^{-1} -z^{10} a^{12}+7 z^6 a^{12}-17 z^4 a^{12}+19 z^2 a^{12}-6 a^{12}-5 z^9 a^{11}+21 z^7 a^{11}-40 z^5 a^{11}+34 z^3 a^{11}-15 z a^{11}+4 a^{11} z^{-1} -z^{10} a^{10}-z^8 a^{10}+13 z^6 a^{10}-29 z^4 a^{10}+18 z^2 a^{10}-5 a^{10}-3 z^9 a^9+11 z^7 a^9-17 z^5 a^9+6 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^8 a^8+8 z^6 a^8-6 z^4 a^8-2 z^2 a^8+a^8-3 z^7 a^7+9 z^5 a^7-9 z^3 a^7+5 z a^7-a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5 }[/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                 4 2     -2 -12               6 2       4 -14             5 5         0 -16           6 5           1 -18         3 5             2 -20       4 6               -2 -22     1 3                 2 -24   1 4                   -3 -26   1                     1 -28 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{ 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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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