L11a184

L11a183

L11a185

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a184 at Knotilus!

Link Presentations

[edit Notes on L11a184's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X18,8,19,7 X10,4,11,3 X22,12,7,11 X16,13,17,14 X12,17,13,18 X6,15,1,16 X4,21,5,22 X2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 6, -8, 7, -3, 9, -7, 8, -4, 11, -2, 10, -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{\left(t(1) t(2)^2-t(2)^2-2 t(1) t(2)+2 t(2)+t(1)-2\right) \left(2 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+2 t(2)+t(1)-1\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-9 q^{3/2}+16 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{25}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^5-a^5 z^3+a^5 z+a^3 z^7+2 a^3 z^5-a^3 z^3-3 a^3 z+a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} +2 a z-z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-a^8 z^4+5 a^7 z^7-9 a^7 z^5+3 a^7 z^3+10 a^6 z^8-22 a^6 z^6+12 a^6 z^4+9 a^5 z^9-12 a^5 z^7-5 a^5 z^5+7 a^5 z^3-2 a^5 z+3 a^4 z^{10}+16 a^4 z^8-47 a^4 z^6+29 a^4 z^4-2 a^4 z^2+17 a^3 z^9-26 a^3 z^7+z^5 a^{-3} +12 a^3 z^3-z^3 a^{-3} -5 a^3 z-a^3 z^{-1} +3 a^2 z^{10}+16 a^2 z^8-39 a^2 z^6+4 z^6 a^{-2} +23 a^2 z^4-5 z^4 a^{-2} -2 a^2 z^2+2 z^2 a^{-2} +a^2+8 a z^9-a z^7+8 z^7 a^{-1} -16 a z^5-11 z^5 a^{-1} +16 a z^3+7 z^3 a^{-1} -5 a z-2 z a^{-1} -a z^{-1} +10 z^8-11 z^6+2 z^4+2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 10 3     -7 0               13 6       7 -2             13 11         -2 -4           14 12           2 -6         10 14             4 -8       7 13               -6 -10     4 10                 6 -12   1 7                   -6 -14   4                     4 -16 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=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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.

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