L11n21

L11n20

L11n22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n21 at Knotilus!

Link Presentations

[edit Notes on L11n21's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,5 X21,14,22,15 X9,18,10,19 X11,20,12,21 X19,10,20,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -8, 10, -9, 4, -6, 7, 11, -2, 3, 8, -10, 9, -7, 6}

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