L11n16

L11n15

L11n17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n16 at Knotilus!

Link Presentations

[edit Notes on L11n16's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 10, -7, 4, -11, -2, 3, 6, -8, 9, -10, 7, -9, 8}

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{2 u v^5-2 u v^4+u v^3+v^2-2 v+2}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+5 z^3 a^{11}+5 z a^{11}+a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-5 z^3 a^9+2 z a^9+2 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-8 z a^7-2 a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{16}+a^{16}-z^3 a^{15}+z a^{15}-z^4 a^{14}+z^2 a^{14}-z^7 a^{13}+4 z^5 a^{13}-4 z^3 a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+11 z^6 a^{12}-19 z^4 a^{12}+13 z^2 a^{12}-3 a^{12}-z^9 a^{11}+4 z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -3 z^8 a^{10}+15 z^6 a^{10}-19 z^4 a^{10}+6 z^2 a^{10}-z^9 a^9+4 z^7 a^9-z^5 a^9-6 z^3 a^9+6 z a^9-2 a^9 z^{-1} -z^8 a^8+4 z^6 a^8-z^4 a^8-5 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+8 z a^7-2 a^7 z^{-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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                   1 1 -8                 1 1 0 -10               2     2 -12           1 1 1     1 -14         1 4 2       1 -16       1 2 1         0 -18       3 3           0 -20   1 2 2             1 -22   1 2               -1 -24 1 1                 0 -26 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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