L11a383

L11a382

L11a384

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a383 at Knotilus!

Link Presentations

[edit Notes on L11a383's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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^4-u^4 v^3-u^3 v^4+4 u^3 v^3-3 u^3 v^2-3 u^2 v^3+7 u^2 v^2-3 u^2 v-3 u v^2+4 u v-u-v+1}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{2}{q^{7/2}}-9 q^{5/2}+\frac{4}{q^{5/2}}+10 q^{3/2}-\frac{7}{q^{3/2}}-q^{13/2}+2 q^{11/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-3} -6 z^5 a^{-3} -12 z^3 a^{-3} -8 z a^{-3} - a^{-3} z^{-1} +z^9 a^{-1} -a z^7+8 z^7 a^{-1} -6 a z^5+24 z^5 a^{-1} -12 a z^3+32 z^3 a^{-1} -8 a z+17 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -5 z^4 a^{-6} +z^2 a^{-6} +3 z^7 a^{-5} -8 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} +a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+22 z^4 a^{-4} +4 a^4 z^2-10 z^2 a^{-4} +3 z^9 a^{-3} +2 a^3 z^7-11 z^7 a^{-3} -7 a^3 z^5+18 z^5 a^{-3} +6 a^3 z^3-11 z^3 a^{-3} -a^3 z+7 z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +2 a^2 z^8+z^8 a^{-2} -5 a^2 z^6-14 z^6 a^{-2} +2 a^2 z^4+31 z^4 a^{-2} -a^2 z^2-16 z^2 a^{-2} + a^{-2} +2 a z^9+5 z^9 a^{-1} -7 a z^7-23 z^7 a^{-1} +15 a z^5+49 z^5 a^{-1} -22 a z^3-47 z^3 a^{-1} +9 a z+19 z a^{-1} - a^{-1} z^{-1} +z^{10}-z^8-3 z^6+10 z^4-10 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     1   -1 10                   3 1   2 8                 4 1     -3 6               5 3       2 4             6 5         -1 2           5 4           1 0         4 7             3 -2       3 4               -1 -4     1 4                 3 -6   1 3                   -2 -8   1                     1 -10 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=-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L11a382 L11a384