L11n417

L11n416

L11n418

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n417 at Knotilus!

Link Presentations

[edit Notes on L11n417's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X15,21,16,20 X5,15,6,14 X13,5,14,4 X19,7,20,12 X11,19,12,18 X17,13,18,22 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 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 w^3-u^2 w^3+u^2 w^2-u^2 w+u^2+u v^2 w^3-u v w^3+u v-u-v^2 w^3+v^2 w^2-v^2 w+v^2-v}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+2 q^6-3 q^5+4 q^4-4 q^3+5 q^2-3 q+4- q^{-1} + q^{-2} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-9 z^2 a^{-2} -2 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-10 a^{-2} +4 a^{-4} +2 a^{-6} - a^{-8} +5-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z a^{-9} +2 z^2 a^{-8} - a^{-8} +z^5 a^{-7} -z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +3 z^6 a^{-6} -10 z^4 a^{-6} +6 z^2 a^{-6} - a^{-6} z^{-2} +5 z^7 a^{-5} -23 z^5 a^{-5} +31 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +4 z^8 a^{-4} -19 z^6 a^{-4} +27 z^4 a^{-4} -21 z^2 a^{-4} -4 a^{-4} z^{-2} +13 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -25 z^5 a^{-3} +50 z^3 a^{-3} -35 z a^{-3} +9 a^{-3} z^{-1} +5 z^8 a^{-2} -29 z^6 a^{-2} +55 z^4 a^{-2} -46 z^2 a^{-2} -5 a^{-2} z^{-2} +22 a^{-2} +z^9 a^{-1} -4 z^7 a^{-1} -z^5 a^{-1} +18 z^3 a^{-1} -19 z a^{-1} +5 a^{-1} z^{-1} +z^8-7 z^6+18 z^4-21 z^2-2 z^{-2} +11 }[/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 χ 15                   1 -1 13                 1   1 11               3 2   -1 9             2 2 1   1 7           3 3       0 5         2 2 1       1 3       2 4           2 1     2 1             1 -1   1 4               3 -3                     0 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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.

L11n416 L11n418