L11a358

L11a357

L11a359

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a358 at Knotilus!

Link Presentations

[edit Notes on L11a358's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,3,15,4 X20,10,21,9 X16,6,17,5 X18,8,19,7 X22,15,11,16 X6,18,7,17 X8,20,9,19 X4,22,5,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 2, -9, 4, -7, 5, -8, 3, -11}, {10, -1, 11, -2, 6, -4, 7, -5, 8, -3, 9, -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(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-3 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{15/2}+3 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} -a z^5+5 z^5 a^{-1} +4 z^5 a^{-3} -z^5 a^{-5} -4 a z^3+8 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} -3 a z+6 z a^{-1} -z a^{-3} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10}-2 a z^9-5 z^9 a^{-1} -3 z^9 a^{-3} -a^2 z^8-2 z^8 a^{-2} -5 z^8 a^{-4} +2 z^8+12 a z^7+24 z^7 a^{-1} +6 z^7 a^{-3} -6 z^7 a^{-5} +6 a^2 z^6+19 z^6 a^{-2} +11 z^6 a^{-4} -6 z^6 a^{-6} +8 z^6-24 a z^5-37 z^5 a^{-1} +2 z^5 a^{-3} +10 z^5 a^{-5} -5 z^5 a^{-7} -11 a^2 z^4-23 z^4 a^{-2} -4 z^4 a^{-4} +7 z^4 a^{-6} -3 z^4 a^{-8} -20 z^4+19 a z^3+25 z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2+7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +10 z^2-5 a z-9 z a^{-1} -3 z a^{-3} -z a^{-7} - a^{-2} + a^{-1} z^{-1} + 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                       1 1 14                     2   -2 12                   3 1   2 10                 4 2     -2 8               5 3       2 6             5 5         0 4           4 4           0 2         4 6             2 0       2 3               -1 -2     1 4                 3 -4   1 2                   -1 -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=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a357 L11a359