L11a123

L11a122

L11a124

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Link Presentations

[edit Notes on L11a123's Link Presentations]

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