L11a51

L11a50

L11a52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a51 at Knotilus!

Link Presentations

[edit Notes on L11a51's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,6,15,5 X10,4,11,3 X20,12,21,11 X22,14,5,13 X12,22,13,21 X2,16,3,15
Gauss code	{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -10, 9, -6, 11, -2, 4, -3, 5, -8, 10, -9}

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{(u-1) (v-1) \left(v^2+1\right) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-11 q^{5/2}+12 q^{3/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^9 a^{-1} -a z^7+7 z^7 a^{-1} -z^7 a^{-3} -5 a z^5+17 z^5 a^{-1} -5 z^5 a^{-3} -7 a z^3+16 z^3 a^{-1} -7 z^3 a^{-3} -2 a z+4 z a^{-1} -2 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10}-4 a z^9-8 z^9 a^{-1} -4 z^9 a^{-3} -4 a^2 z^8+4 z^8 a^{-2} -4 z^8 a^{-4} +4 z^8-3 a^3 z^7+15 a z^7+36 z^7 a^{-1} +14 z^7 a^{-3} -4 z^7 a^{-5} -a^4 z^6+13 a^2 z^6+9 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6+10 a^3 z^5-21 a z^5-65 z^5 a^{-1} -24 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-9 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} +7 z^4 a^{-6} -10 z^4-6 a^3 z^3+16 a z^3+44 z^3 a^{-1} +16 z^3 a^{-3} -4 z^3 a^{-5} +2 z^3 a^{-7} -a^4 z^2+a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +5 z^2-4 a z-8 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   3 1   2 8                 5 2     -3 6               6 3       3 4             6 5         -1 2           7 6           1 0         5 8             3 -2       3 5               -2 -4     2 5                 3 -6   1 3                   -2 -8   2                     2 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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.

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