L11a12

L11a11

L11a13

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Link Presentations

[edit Notes on L11a12's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,12,15,11 X8493 X12,5,13,6 X22,13,5,14 X18,15,19,16 X16,9,17,10 X10,17,11,18 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}

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{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{21}{q^{5/2}}-\frac{25}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z-3 a^7 z^3-4 a^7 z-a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+6 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-2 a^3 z+a z^5+a z^3-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-a^{11} z^3+4 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+7 a^9 z^7-9 a^9 z^5+5 a^9 z^3-2 a^9 z+8 a^8 z^8-5 a^8 z^6-7 a^8 z^4+9 a^8 z^2-2 a^8+6 a^7 z^9+5 a^7 z^7-25 a^7 z^5+24 a^7 z^3-8 a^7 z+a^7 z^{-1} +2 a^6 z^{10}+17 a^6 z^8-36 a^6 z^6+15 a^6 z^4+8 a^6 z^2-5 a^6+13 a^5 z^9-10 a^5 z^7-26 a^5 z^5+31 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+18 a^4 z^8-47 a^4 z^6+27 a^4 z^4-3 a^4+7 a^3 z^9-3 a^3 z^7-21 a^3 z^5+18 a^3 z^3-4 a^3 z+9 a^2 z^8-19 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-10 a z^5+5 a z^3+a z-a z^{-1} +z^6-z^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     4   4 0                   6 1   -5 -2                 11 4     7 -4               12 8       -4 -6             13 9         4 -8           11 12           1 -10         10 13             -3 -12       5 11               6 -14     3 10                 -7 -16   1 5                   4 -18   3                     -3 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/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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