L11a11

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Link Presentations

[edit Notes on L11a11's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X12,5,13,6 X22,13,5,14 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 9, -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{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{16}{q^{9/2}}-q^{7/2}+\frac{23}{q^{7/2}}+5 q^{5/2}-\frac{27}{q^{5/2}}-12 q^{3/2}+\frac{27}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+18 \sqrt{q}-\frac{25}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+3 a^5 z^3+3 a^5 z-3 a^3 z^5-6 a^3 z^3-4 a^3 z+a z^7+3 a z^5-z^5 a^{-1} +5 a z^3-z^3 a^{-1} +3 a z+a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+7 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+10 a^6 z^4-3 a^6 z^2+6 a^5 z^9-22 a^5 z^5+23 a^5 z^3-6 a^5 z+2 a^4 z^{10}+20 a^4 z^8-53 a^4 z^6+41 a^4 z^4-10 a^4 z^2+15 a^3 z^9-9 a^3 z^7-34 a^3 z^5+z^5 a^{-3} +35 a^3 z^3-8 a^3 z+2 a^2 z^{10}+28 a^2 z^8-64 a^2 z^6+5 z^6 a^{-2} +41 a^2 z^4-3 z^4 a^{-2} -9 a^2 z^2+9 a z^9+7 a z^7+12 z^7 a^{-1} -38 a z^5-16 z^5 a^{-1} +26 a z^3+7 z^3 a^{-1} -6 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +15 z^8-21 z^6+9 z^4-3 z^2-1 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 10 4     -6 0               15 8       7 -2             14 12         -2 -4           13 13           0 -6         10 14             4 -8       6 13               -7 -10     3 10                 7 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/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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