L11n219

L11n218

L11n220

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n219 at Knotilus!

Link Presentations

[edit Notes on L11n219's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X4,22,5,21 X14,5,15,6 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, -6, 7, 10, -11, -2}, {2, -1, 9, 3, -8, -7, 4, -9, -5, 11, -10, 8, 6, -4}

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