L11a201

L11a200

L11a202

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a201 at Knotilus!

Link Presentations

[edit Notes on L11a201's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X22,18,7,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 8, -9, 11, -4, 5, -6, 9, -8, 7, -5}

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{2 t(1)^2 t(2)^4-5 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-5 t(2)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{7/2}-2 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^7-a z^7+a^5 z^5-4 a^3 z^5-5 a z^5+z^5 a^{-1} +3 a^5 z^3-4 a^3 z^3-9 a z^3+4 z^3 a^{-1} +2 a^5 z-a^3 z-8 a z+4 z a^{-1} +a^5 z^{-1} -2 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3+3 a^8 z^4+6 a^7 z^5-5 a^7 z^3+2 a^7 z+8 a^6 z^6-10 a^6 z^4+2 a^6 z^2+9 a^5 z^7-18 a^5 z^5+10 a^5 z^3-5 a^5 z+a^5 z^{-1} +7 a^4 z^8-14 a^4 z^6+a^4 z^4+3 a^4 z^2-a^4+4 a^3 z^9-6 a^3 z^7-9 a^3 z^5+9 a^3 z^3-a^3 z+a^2 z^{10}+6 a^2 z^8+z^8 a^{-2} -34 a^2 z^6-6 z^6 a^{-2} +41 a^2 z^4+12 z^4 a^{-2} -16 a^2 z^2-9 z^2 a^{-2} +3 a^2+2 a^{-2} +6 a z^9+2 z^9 a^{-1} -26 a z^7-11 z^7 a^{-1} +35 a z^5+20 z^5 a^{-1} -21 a z^3-14 z^3 a^{-1} +10 a z+4 z a^{-1} -2 a z^{-1} - a^{-1} z^{-1} +z^{10}-18 z^6+39 z^4-26 z^2+5 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     1   1 4                   3 1   -2 2                 4 1     3 0               5 3       -2 -2             6 4         2 -4           6 6           0 -6         5 5             0 -8       3 6               3 -10     3 5                 -2 -12     3                   3 -14 1 3                     -2 -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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