L10a82

L10a81

L10a83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a82 at Knotilus!

Link Presentations

[edit Notes on L10a82's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,10,7,9 X10,14,11,13 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -8, 2, -9, 5, -10}, {8, -1, 3, -4, 9, -2, 4, -7, 6, -5, 10, -6, 7, -3}

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)^2-5 t(2) t(1)^2+3 t(1)^2-5 t(2)^2 t(1)+9 t(2) t(1)-5 t(1)+3 t(2)^2-5 t(2)+2}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -7 q^{9/2}+10 q^{7/2}-\frac{1}{q^{7/2}}-13 q^{5/2}+\frac{2}{q^{5/2}}+13 q^{3/2}-\frac{6}{q^{3/2}}-q^{13/2}+4 q^{11/2}-12 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+z a^{-3} +a^3 z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +6 z^7 a^{-5} -11 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} -11 z^4 a^{-4} +6 z^2 a^{-4} +z^9 a^{-3} +11 z^7 a^{-3} +a^3 z^5-27 z^5 a^{-3} -3 a^3 z^3+21 z^3 a^{-3} +3 a^3 z-5 z a^{-3} -a^3 z^{-1} +7 z^8 a^{-2} +2 a^2 z^6-8 z^6 a^{-2} -3 a^2 z^4+2 z^2 a^{-2} +a^2+z^9 a^{-1} +3 a z^7+8 z^7 a^{-1} -3 a z^5-19 z^5 a^{-1} -a z^3+17 z^3 a^{-1} +2 a z-5 z a^{-1} -a z^{-1} +3 z^8-2 z^6+z^4-2 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 4 1   3 8               6 3     -3 6             7 4       3 4           6 6         0 2         6 7           -1 0       4 7             3 -2     2 5               -3 -4   1 5                 4 -6   1                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

L10a81 L10a83