L11a24

L11a23

L11a25

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Link Presentations

[edit Notes on L11a24's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X22,19,5,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -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{3 u v^4-10 u v^3+12 u v^2-7 u v+u+v^5-7 v^4+12 v^3-10 v^2+3 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{21}{q^{9/2}}-\frac{22}{q^{11/2}}+\frac{18}{q^{13/2}}-\frac{14}{q^{15/2}}+\frac{8}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -6 a^7 z^3-9 a^7 z-3 a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+7 a^5 z+2 a^5 z^{-1} +a^3 z^5-a^3 z^3-3 a^3 z-a^3 z^{-1} -a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+7 z^5 a^{11}-6 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -5 z^8 a^{10}+9 z^6 a^{10}-3 z^4 a^{10}-2 z^2 a^{10}+2 a^{10}-4 z^9 a^9-2 z^7 a^9+21 z^5 a^9-21 z^3 a^9+11 z a^9-3 a^9 z^{-1} -z^{10} a^8-16 z^8 a^8+42 z^6 a^8-34 z^4 a^8+12 z^2 a^8-9 z^9 a^7+2 z^7 a^7+31 z^5 a^7-36 z^3 a^7+15 z a^7-3 a^7 z^{-1} -z^{10} a^6-19 z^8 a^6+48 z^6 a^6-45 z^4 a^6+18 z^2 a^6-2 a^6-5 z^9 a^5-5 z^7 a^5+27 z^5 a^5-31 z^3 a^5+13 z a^5-2 a^5 z^{-1} -8 z^8 a^4+13 z^6 a^4-13 z^4 a^4+6 z^2 a^4-6 z^7 a^3+9 z^5 a^3-8 z^3 a^3+5 z a^3-a^3 z^{-1} -3 z^6 a^2+4 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a }[/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 2                       1 1 0                     2   -2 -2                   6 1   5 -4                 9 3     -6 -6               10 5       5 -8             11 9         -2 -10           11 10           1 -12         8 12             4 -14       6 10               -4 -16     2 8                 6 -18   1 6                   -5 -20   2                     2 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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