L11n161

L11n160

L11n162

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n161 at Knotilus!

Link Presentations

[edit Notes on L11n161's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X5,13,6,12 X6,21,1,22 X13,20,14,21 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}

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 t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{7}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{6}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+a^7 z-a^7 z^{-1} -a^5 z^5-2 a^5 z^3+a^5 z+3 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-4 a^3 z-2 a^3 z^{-1} +a z^3+a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-4 a^9 z^5+4 a^9 z^3-a^9 z+3 a^8 z^8-14 a^8 z^6+18 a^8 z^4-6 a^8 z^2-a^8+2 a^7 z^9-6 a^7 z^7-a^7 z^5+7 a^7 z^3-2 a^7 z+a^7 z^{-1} +7 a^6 z^8-30 a^6 z^6+34 a^6 z^4-9 a^6 z^2-3 a^6+2 a^5 z^9-4 a^5 z^7-6 a^5 z^5+10 a^5 z^3-5 a^5 z+3 a^5 z^{-1} +4 a^4 z^8-15 a^4 z^6+16 a^4 z^4-4 a^4 z^2-3 a^4+3 a^3 z^7-9 a^3 z^5+10 a^3 z^3-6 a^3 z+2 a^3 z^{-1} +a^2 z^6+3 a z^3-2 a z+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   1 -1 0                 2   2 -2               3 2   -1 -4             4 1     3 -6           3 4       1 -8         4 3         1 -10       2 3           1 -12     2 4             -2 -14   1 2               1 -16   2                 -2 -18 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=-8 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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.

L11n160 L11n162