L11n280

L11n279

L11n281

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n280 at Knotilus!

Link Presentations

[edit Notes on L11n280's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {-8, 2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

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{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^3+2 q^2-5 q+8-7 q^{-1} +9 q^{-2} -6 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^6+a^4 z^4-4 a^2 z^4+2 z^4+2 a^4 z^2-7 a^2 z^2-z^2 a^{-2} +6 z^2+2 a^4-8 a^2-2 a^{-2} +8+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+5 a^5 z^3+3 a^4 z^8-7 a^4 z^6-a^4 z^2-2 a^4 z^{-2} +4 a^4+a^3 z^9+3 a^3 z^7-17 a^3 z^5+18 a^3 z^3+z^3 a^{-3} -12 a^3 z-2 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +5 a^2 z^8-15 a^2 z^6+15 a^2 z^4+2 z^4 a^{-2} -11 a^2 z^2-2 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +10 a^2+ a^{-2} +a z^9+a z^7+z^7 a^{-1} -11 a z^5-3 z^5 a^{-1} +23 a z^3+11 z^3 a^{-1} -21 a z-11 z a^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} +2 z^8-7 z^6+14 z^4-9 z^2-4 z^{-2} +7 }[/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 χ 7                   1 -1 5                 1   1 3               4 1   -3 1             4 1     3 -1           5 6       1 -3         4 3 1       2 -5       2 5           3 -7     4 4             0 -9   1 4               3 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

L11n279 L11n281