L11n1

L11a548

L11n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n1 at Knotilus!

Link L11n1.

A graph, L11n1.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -11, -2, 3, 10, -7, 8, -9, 6, -8, 7}

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{t(1) t(2)^5-4 t(1) t(2)^4+2 t(1) t(2)^3+2 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +a^9 z^3+5 a^9 z+3 a^9 z^{-1} -2 a^7 z^5-9 a^7 z^3-9 a^7 z-3 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{12}+a^{12}-2 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^6 a^{10}+3 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}-2 z^7 a^9+9 z^5 a^9-16 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^8 a^8+8 z^6 a^8-9 z^4 a^8+4 z^2 a^8-z^9 a^7+z^7 a^7+12 z^5 a^7-25 z^3 a^7+17 z a^7-3 a^7 z^{-1} -4 z^8 a^6+19 z^6 a^6-25 z^4 a^6+12 z^2 a^6-2 a^6-z^9 a^5+2 z^7 a^5+8 z^5 a^5-18 z^3 a^5+11 z a^5-2 a^5 z^{-1} -2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+4 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+4 z a^3-a^3 z^{-1} }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               2 1   1 -6           1 3 2     0 -8           3 1       2 -10       1 3 3         1 -12     1 4 3           0 -14     1 2             1 -16   2 3               -1 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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.

L11a548 L11n2