L11n355

L11n354

L11n356

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n355 at Knotilus!

Link Presentations

[edit Notes on L11n355's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,16,8,17 X15,8,16,9 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}

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-u v^2-2 u v w+3 u v+u w-u-v^2 w+v^2+3 v w-2 v-w+1\right)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-7} -4 q^{-6} +9 q^{-5} -10 q^{-4} +13 q^{-3} -q^2-12 q^{-2} +4 q+10 q^{-1} -7 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +z^2 a^6-2 a^6 z^{-2} -2 a^6-2 z^4 a^4-2 z^2 a^4+a^4 z^{-2} +a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +4 a^8+a^7 z^7+a^7 z^5+a^7 z^3-5 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-2 a^6 z^6+7 a^6 z^4-10 a^6 z^2-2 a^6 z^{-2} +6 a^6+a^5 z^9+4 a^5 z^7-7 a^5 z^5+7 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +6 a^4 z^8-7 a^4 z^6+2 a^4 z^4-a^4 z^2-a^4 z^{-2} +2 a^4+a^3 z^9+9 a^3 z^7-20 a^3 z^5+11 a^3 z^3+4 a^2 z^8-a^2 z^6-9 a^2 z^4+5 a^2 z^2-a^2+6 a z^7-11 a z^5+z^5 a^{-1} +4 a z^3-z^3 a^{-1} +4 z^6-7 z^4+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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             6 3     3 -3           7 5       -2 -5         6 5         1 -7       5 8           3 -9     4 5             -1 -11   1 6               5 -13 1 3                 -2 -15 2                   2

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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L11n354 L11n356