L11n422

L11n421

L11n423

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n422 at Knotilus!

Link Presentations

[edit Notes on L11n422's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X19,5,20,4 X12,22,7,21
Gauss code	{1, -4, -3, 10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, 4, 5, -10, -6, 11, -8}

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^2 v^2 w^2-2 u^2 v^2 w-u^2 v w^2+2 u^2 v w-u^2 v-u v^2 w^2+u v^2 w+3 u v w^2-3 u v w-u w^2+u w+v w^3-2 v w^2+v w+2 w^2-w}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-2 q^8+5 q^7-6 q^6+8 q^5-8 q^4+8 q^3-5 q^2+4 q-1 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +6 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +3 a^{-4} -6 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} +2 z^7 a^{-9} -5 z^6 a^{-4} -17 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +z^5 a^{-3} -8 z^5 a^{-5} -15 z^5 a^{-7} -6 z^5 a^{-9} +4 z^4 a^{-2} +11 z^4 a^{-4} +25 z^4 a^{-6} +14 z^4 a^{-8} -4 z^4 a^{-10} +z^3 a^{-1} +14 z^3 a^{-5} +18 z^3 a^{-7} +3 z^3 a^{-9} -3 z^2 a^{-2} -10 z^2 a^{-4} -23 z^2 a^{-6} -12 z^2 a^{-8} +4 z^2 a^{-10} -9 z a^{-5} -9 z a^{-7} - a^{-2} +5 a^{-4} +11 a^{-6} +5 a^{-8} - a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} 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   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 2 1 -1 15               3     3 13             3 2     -1 11           5 3       2 9         3 3         0 7       5 5           0 5     2 5             3 3   2 3               -1 1   3                 3 -1 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{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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