L11n12

L11n11

L11n13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n12 at Knotilus!

Link Presentations

[edit Notes on L11n12's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X8493 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -7, 8, -6, 9, 11, -2, 3, 10, -9, 7, -8, 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{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-2 a^9 z-a^9 z^{-1} +a^7 z^5+4 a^7 z^3+5 a^7 z+a^7 z^{-1} -a^5 z^3+2 a^5 z^{-1} -a^3 z^3-3 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2-a^{12}+2 a^{11} z^7-8 a^{11} z^5+6 a^{11} z^3-a^{11} z+2 a^{10} z^8-8 a^{10} z^6+6 a^{10} z^4+a^9 z^9-3 a^9 z^7-a^9 z^5+4 a^9 z^3-a^9 z^{-1} +3 a^8 z^8-15 a^8 z^6+23 a^8 z^4-13 a^8 z^2+3 a^8+a^7 z^9-5 a^7 z^7+8 a^7 z^5-5 a^7 z^3+4 a^7 z-a^7 z^{-1} +a^6 z^8-6 a^6 z^6+13 a^6 z^4-8 a^6 z^2+a^5 z^5-2 a^5 z^3+2 a^5 z^{-1} +2 a^4 z^2-3 a^4+a^3 z^3-3 a^3 z+2 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                   1 1 -4                 1 2 1 -6               2   1 3 -8             2 2     0 -10           3 2       1 -12         2 3 1       0 -14       2 3           -1 -16     1 2 1           0 -18   1 2               -1 -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=-6 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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}^{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.

L11n11 L11n13