L11n128

L11n127

L11n129

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n128 at Knotilus!

Link Presentations

[edit Notes on L11n128's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X16,12,17,11 X6,18,1,17 X19,14,20,15 X13,20,14,21 X21,19,22,18
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 11, -9, 10, -11, -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{u^2 v^4-3 u^2 v^3+2 u^2 v^2-u v^4+2 u v^3-u v^2+2 u v-u+2 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 4 q^{9/2}-6 q^{7/2}+6 q^{5/2}-\frac{1}{q^{5/2}}-6 q^{3/2}+\frac{2}{q^{3/2}}+q^{13/2}-3 q^{11/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-8 z^3 a^{-1} +8 z^3 a^{-3} -4 z^3 a^{-5} +3 a z-7 z a^{-1} +7 z a^{-3} -3 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -3 z^8 a^{-4} -2 z^8-a z^7-2 z^7 a^{-3} -3 z^7 a^{-5} +20 z^6 a^{-2} +10 z^6 a^{-4} -z^6 a^{-6} +9 z^6+5 a z^5+15 z^5 a^{-1} +20 z^5 a^{-3} +10 z^5 a^{-5} -20 z^4 a^{-2} -9 z^4 a^{-4} -11 z^4-8 a z^3-25 z^3 a^{-1} -27 z^3 a^{-3} -13 z^3 a^{-5} -3 z^3 a^{-7} +6 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-8} +4 z^2+5 a z+13 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 2   2 10               2 1   -1 8             4 2     2 6           3 3       0 4         3 3         0 2       3 4           1 0     1 2             -1 -2   1 3               2 -4   1                 -1 -6 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=4 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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.

L11n127 L11n129