L11a364

L11a363

L11a365

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a364 at Knotilus!

Link Presentations

[edit Notes on L11a364's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X16,6,17,5 X22,16,11,15 X18,8,19,7 X20,10,21,9 X6,18,7,17 X8,20,9,19 X10,22,1,21
Gauss code	{1, -2, 3, -4, 5, -9, 7, -10, 8, -11}, {4, -1, 2, -3, 6, -5, 9, -7, 10, -8, 11, -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)^4 t(1)^4-t(2)^3 t(1)^4-t(2)^4 t(1)^3+3 t(2)^3 t(1)^3-2 t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-2 q^{15/2}+3 q^{13/2}-5 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-5} +6 z^5 a^{-5} +11 z^3 a^{-5} +6 z a^{-5} -z^9 a^{-3} -8 z^7 a^{-3} -23 z^5 a^{-3} -28 z^3 a^{-3} -12 z a^{-3} - a^{-3} z^{-1} +z^7 a^{-1} +6 z^5 a^{-1} +11 z^3 a^{-1} +7 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -4 z^9 a^{-3} -2 z^9 a^{-5} +3 z^8 a^{-2} +3 z^8 a^{-4} -2 z^8 a^{-6} -2 z^8-a z^7+10 z^7 a^{-1} +22 z^7 a^{-3} +9 z^7 a^{-5} -2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +6 z^6 a^{-6} -2 z^6 a^{-8} +10 z^6+5 a z^5-16 z^5 a^{-1} -45 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4 a^{-6} +3 z^4 a^{-8} -z^4 a^{-10} -13 z^4-6 a z^3+14 z^3 a^{-1} +43 z^3 a^{-3} +15 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +5 z^2 a^{-2} +5 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} +4 z^2+a z-7 z a^{-1} -14 z a^{-3} -5 z a^{-5} -z a^{-9} - 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     1   1 14                   2 1   -1 12                 3 1     2 10               3 2       -1 8             4 3         1 6           3 4           1 4         3 3             0 2       2 4               2 0     1 2                 -1 -2   1 2                   1 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/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=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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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.

L11a363 L11a365