L11a35

L11a34

L11a36

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a35 at Knotilus!

Link Presentations

[edit Notes on L11a35's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X18,13,19,14 X14,17,15,18 X22,20,5,19 X20,12,21,11 X12,22,13,21 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 7, -8, 4, -5, 9, -3, 5, -4, 6, -7, 8, -6}

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(2)^5+2 t(1) t(2)^4-6 t(2)^4-7 t(1) t(2)^3+9 t(2)^3+9 t(1) t(2)^2-7 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{3}{q^{9/2}}+6 q^{7/2}-\frac{7}{q^{7/2}}-10 q^{5/2}+\frac{10}{q^{5/2}}+14 q^{3/2}-\frac{14}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} +z a^{-5} -3 a^3 z^3-2 z^3 a^{-3} -5 a^3 z-2 a^3 z^{-1} -z a^{-3} + a^{-3} z^{-1} +2 a z^5+z^5 a^{-1} +5 a z^3-z^3 a^{-1} +6 a z+3 a z^{-1} -4 z a^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-4 a^5 z^5+3 z^5 a^{-5} +6 a^5 z^3-3 z^3 a^{-5} -4 a^5 z+z a^{-5} +a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-5 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +3 a^3 z^9-5 a^3 z^7+6 z^7 a^{-3} -11 a^3 z^5-5 z^5 a^{-3} +22 a^3 z^3+z^3 a^{-3} -11 a^3 z-z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -36 a^2 z^6-5 z^6 a^{-2} +38 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+8 z^2 a^{-2} +2 a^2-2 a^{-2} +7 a z^9+4 z^9 a^{-1} -11 a z^7+z^7 a^{-1} -18 a z^5-19 z^5 a^{-1} +35 a z^3+23 z^3 a^{-1} -18 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +z^{10}+11 z^8-35 z^6+27 z^4-6 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 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 6 2     4 4               8 4       -4 2             8 6         2 0           8 9           1 -2         6 7             -1 -4       4 8               4 -6     3 6                 -3 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{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.

L11a34 L11a36