L11a522

L11a521

L11a523

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a522 at Knotilus!

Link Presentations

[edit Notes on L11a522's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X16,5,17,6 X18,8,19,7 X12,20,7,19 X10,14,11,13 X22,10,13,9 X20,15,21,16 X6,17,1,18 X2,11,3,12 X4,22,5,21
Gauss code	{1, -10, 2, -11, 3, -9}, {4, -1, 7, -6, 10, -5}, {6, -2, 8, -3, 9, -4, 5, -8, 11, -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{u^2 v^2 w^3-2 u^2 v^2 w^2+u^2 v^2 w-2 u^2 v w^3+4 u^2 v w^2-3 u^2 v w+u^2 v+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-4 u v^2 w+u v^2+3 u v w^3-8 u v w^2+8 u v w-3 u v-u w^3+4 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2-v w^3+3 v w^2-4 v w+2 v-w^2+2 w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+10 q^4-15 q^3+22 q^2-24 q+25-21 q^{-1} +16 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-9 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^8+5 z^6+10 z^4+8 z^2+ z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+9 a z^9+17 z^9 a^{-1} +8 z^9 a^{-3} +11 a^2 z^8+13 z^8 a^{-2} +8 z^8 a^{-4} +16 z^8+8 a^3 z^7-10 a z^7-36 z^7 a^{-1} -14 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-52 z^6 a^{-2} -19 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-11 a^3 z^5-2 a z^5+21 z^5 a^{-1} +3 z^5 a^{-3} -8 z^5 a^{-5} -5 a^4 z^4+15 a^2 z^4+62 z^4 a^{-2} +16 z^4 a^{-4} -2 z^4 a^{-6} +64 z^4-a^5 z^3+4 a^3 z^3+9 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +3 z^3 a^{-5} +a^4 z^2-7 a^2 z^2-37 z^2 a^{-2} -12 z^2 a^{-4} +z^2 a^{-6} -32 z^2-a^3 z-3 a z-8 z a^{-1} -6 z a^{-3} +a^2+12 a^{-2} +6 a^{-4} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   7 1   6 7                 9 4     -5 5               13 6       7 3             12 10         -2 1           13 12           1 -1         9 13             4 -3       7 12               -5 -5     3 10                 7 -7   1 6                   -5 -9   3                     3 -11 1                       -1

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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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