L11n401

L11n400

L11n402

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n401 at Knotilus!

Link Presentations

[edit Notes on L11n401's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X18,14,5,13 X14,12,15,11 X12,18,13,17 X16,22,17,21 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3}, {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, 3, -9, 8, -6}

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{(v-1) (w-1) \left(u w^2-2 u w+2 u-2 w^2+2 w-1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4+3 q^{-4} -7 q^3-5 q^{-3} +11 q^2+11 q^{-2} -13 q-12 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-4} +2 a^4 z^{-2} +2 a^4- a^{-4} +a^2 z^4+2 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -7 a^2+ a^{-2} -z^6-2 z^4+4 z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a z^9+z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+3 a^3 z^7+11 a z^7+13 z^7 a^{-1} +5 z^7 a^{-3} -8 a^2 z^6+3 z^6 a^{-4} -11 z^6-6 a^3 z^5-32 a z^5-35 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} +6 a^4 z^4+18 a^2 z^4-10 z^4 a^{-2} -5 z^4 a^{-4} +7 z^4+14 a^3 z^3+43 a z^3+36 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} -13 a^4 z^2-26 a^2 z^2+6 z^2 a^{-2} +3 z^2 a^{-4} -10 z^2-16 a^3 z-32 a z-20 z a^{-1} -3 z a^{-3} +z a^{-5} +9 a^4+18 a^2- a^{-4} +11+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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 χ 11                   1 -1 9                 2   2 7               5 1   -4 5             6 2     4 3           7 5       -2 1         7 6         1 -1       7 9           2 -3     4 5             -1 -5   1 7               6 -7 2 4                 -2 -9 3                   3

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=1 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_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.

L11n400 L11n402