L11n303

L11n302

L11n304

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n303 at Knotilus!

Link Presentations

[edit Notes on L11n303's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}

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{-t(2)^2 t(3)^4+t(1) t(2)^2 t(3)^3-t(1) t(2) t(3)^3+t(2) t(3)^3-t(1) t(2)^2 t(3)^2+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2+t(3)^2-t(1) t(2) t(3)+t(2) t(3)-t(3)+t(1)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+2 q^7-3 q^6+4 q^5-4 q^4+4 q^3-2 q^2+2 q+1+ q^{-2} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} +z^6 a^{-4} -8 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +z^4-19 z^2 a^{-2} +14 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-17 a^{-2} +13 a^{-4} -3 a^{-6} +7-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +z^8 a^{-6} +z^8+z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +2 z^7 a^{-7} -10 z^6 a^{-2} -6 z^6 a^{-4} -2 z^6 a^{-6} +2 z^6 a^{-8} -8 z^6-9 z^5 a^{-1} -16 z^5 a^{-3} -14 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +31 z^4 a^{-2} +15 z^4 a^{-4} -z^4 a^{-6} -6 z^4 a^{-8} +21 z^4+22 z^3 a^{-1} +41 z^3 a^{-3} +26 z^3 a^{-5} +4 z^3 a^{-7} -3 z^3 a^{-9} -39 z^2 a^{-2} -16 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} -23 z^2-19 z a^{-1} -35 z a^{-3} -19 z a^{-5} -2 z a^{-7} +z a^{-9} +22 a^{-2} +13 a^{-4} - a^{-8} +11+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     1   1 13                   2 1   -1 11                 2 1     1 9               2 2       0 7           1 3 2         0 5           2 4           2 3       1 2 1             0 1         3               3 -1     1                   1 -3 1                       1 -5 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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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