L11n373

L11n372

L11n374

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n373 at Knotilus!

Link Presentations

[edit Notes on L11n373's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,19,8,18 X21,15,22,14 X9,20,10,21 X13,9,14,8 X15,17,16,22 X17,5,18,16 X19,12,20,13 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {-8, 3, -9, 5, -4, 7}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 4, -7, 8}

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{(w-1) \left(u v^2 w^3-u v^2 w^2-u v w^3+2 u v w^2-2 u v w-u w^2+u w+v^2 w^2-v^2 w-2 v w^2+2 v w-v-w+1\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4+3 q^{-4} -6 q^3-5 q^{-3} +10 q^2+9 q^{-2} -11 q-11 q^{-1} +13 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^6-z^6 a^{-2} -5 a^2 z^4-4 z^4 a^{-2} -10 a^2 z^2-5 z^2 a^{-2} -2 a^2 z^{-2} -8 a^2- a^{-2} +z^8+6 z^6+13 z^4+12 z^2+ z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +5 a^2 z^8+4 z^8 a^{-2} +9 z^8+3 a^3 z^7+a z^7+2 z^7 a^{-1} +4 z^7 a^{-3} -18 a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} -26 z^6-6 a^3 z^5-10 a z^5-10 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +6 a^4 z^4+37 a^2 z^4-6 z^4 a^{-4} +37 z^4+7 a^3 z^3+18 a z^3+11 z^3 a^{-1} -2 z^3 a^{-3} -2 z^3 a^{-5} -14 a^4 z^2-36 a^2 z^2+z^2 a^{-2} +3 z^2 a^{-4} -24 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 χ 11                   1 -1 9                 2   2 7               4 1   -3 5             6 2     4 3           6 5       -1 1         7 5         2 -1       5 7           2 -3     4 6             -2 -5   2 6               4 -7 1 3                 -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} }[/math] [math]\displaystyle{ r=-3 }[/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}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

L11n372 L11n374