L11a384

L11a383

L11a385

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a384 at Knotilus!

Link Presentations

[edit Notes on L11a384's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,8,15,7 X16,4,17,3 X20,6,21,5 X18,10,19,9 X10,18,1,17 X4,20,5,19 X22,14,11,13 X6,16,7,15 X2,11,3,12 X8,22,9,21
Gauss code	{1, -10, 3, -7, 4, -9, 2, -11, 5, -6}, {10, -1, 8, -2, 9, -3, 6, -5, 7, -4, 11, -8}

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^4 v^2-u^4 v+3 u^3 v^3-8 u^3 v^2+5 u^3 v-u^3+u^2 v^4-8 u^2 v^3+13 u^2 v^2-8 u^2 v+u^2-u v^4+5 u v^3-8 u v^2+3 u v-v^3+v^2}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 22 q^{9/2}-19 q^{7/2}+14 q^{5/2}-8 q^{3/2}+q^{21/2}-4 q^{19/2}+9 q^{17/2}-15 q^{15/2}+19 q^{13/2}-23 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} -z^5 a^{-7} +z^3 a^{-7} +2 z a^{-7} - a^{-7} z^{-1} -3 z^5 a^{-5} -6 z^3 a^{-5} -3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} +z^3 a^{-3} +3 z a^{-3} +z^3 a^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-6} -2 z^{10} a^{-8} -6 z^9 a^{-5} -12 z^9 a^{-7} -6 z^9 a^{-9} -8 z^8 a^{-4} -13 z^8 a^{-6} -12 z^8 a^{-8} -7 z^8 a^{-10} -6 z^7 a^{-3} +2 z^7 a^{-5} +18 z^7 a^{-7} +6 z^7 a^{-9} -4 z^7 a^{-11} -3 z^6 a^{-2} +13 z^6 a^{-4} +33 z^6 a^{-6} +33 z^6 a^{-8} +15 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +9 z^5 a^{-3} +8 z^5 a^{-5} -4 z^5 a^{-7} +7 z^5 a^{-9} +9 z^5 a^{-11} +4 z^4 a^{-2} -13 z^4 a^{-4} -28 z^4 a^{-6} -21 z^4 a^{-8} -8 z^4 a^{-10} +2 z^4 a^{-12} +2 z^3 a^{-1} -7 z^3 a^{-3} -8 z^3 a^{-5} +2 z^3 a^{-7} -5 z^3 a^{-9} -6 z^3 a^{-11} -z^2 a^{-2} +6 z^2 a^{-4} +10 z^2 a^{-6} +5 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -z a^{-1} +3 z a^{-3} -z a^{-5} -5 z a^{-7} +z a^{-9} +z a^{-11} - a^{-6} + a^{-5} z^{-1} + a^{-7} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   6 1   -5 16                 9 3     6 14               11 7       -4 12             12 8         4 10           10 11           1 8         9 12             -3 6       5 10               5 4     3 9                 -6 2   1 6                   5 0   2                     -2 -2 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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.

L11a383 L11a385