L11n33

L11n32

L11n34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n33 at Knotilus!

Link Presentations

[edit Notes on L11n33's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}

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-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{9/2}+\frac{1}{q^{9/2}}-3 q^{7/2}-\frac{2}{q^{7/2}}+3 q^{5/2}+\frac{1}{q^{5/2}}-2 q^{3/2}-\frac{1}{q^{3/2}}-q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^5-z^5 a^{-1} -a^3 z^3+5 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} -2 a^3 z+6 a z-8 z a^{-1} +5 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-5} -5 z^5 a^{-5} +7 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} +a^4 z^6-10 z^6 a^{-4} -4 a^4 z^4+13 z^4 a^{-4} +2 a^4 z^2-7 z^2 a^{-4} + a^{-4} +z^9 a^{-3} +2 a^3 z^7-2 z^7 a^{-3} -10 a^3 z^5-11 z^5 a^{-3} +11 a^3 z^3+24 z^3 a^{-3} -5 a^3 z-16 z a^{-3} +3 a^{-3} z^{-1} +a^2 z^8+4 z^8 a^{-2} -5 a^2 z^6-23 z^6 a^{-2} +3 a^2 z^4+34 z^4 a^{-2} -17 z^2 a^{-2} +a^2+3 a^{-2} +z^9 a^{-1} +3 a z^7-2 z^7 a^{-1} -21 a z^5-17 z^5 a^{-1} +34 a z^3+40 z^3 a^{-1} -17 a z-24 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +3 z^8-19 z^6+28 z^4-12 z^2+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 12                       1 1 10                     1   -1 8                   2 1   1 6               1 2 1     0 4             1 2 2       -1 2           2 3 2         1 0         2 4 2           0 -2       1 2 3             2 -4     1 2 1               0 -6   1 1 1                 1 -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=-2 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

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