L11n332

L11n331

L11n333

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n332 at Knotilus!

Link Presentations

[edit Notes on L11n332's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X3849 X17,2,18,3 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code	{1, 4, -3, -11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, -4, -6, 11, 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{(v+w-1) (v w-v-w) (u v w-1)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-9} +2 q^{-8} -3 q^{-7} +7 q^{-6} -5 q^{-5} +6 q^{-4} -5 q^{-3} +4 q^{-2} -2 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+z^4 a^8+4 z^2 a^8+a^8 z^{-2} +2 a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-2 a^6 z^{-2} -3 a^6-z^6 a^4-3 z^4 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{11}-2 z a^{11}+2 z^4 a^{10}-4 z^2 a^{10}+3 a^{10}+z^7 a^9-4 z^5 a^9+10 z^3 a^9-7 z a^9+2 z^8 a^8-10 z^6 a^8+22 z^4 a^8-19 z^2 a^8-a^8 z^{-2} +11 a^8+z^9 a^7-2 z^7 a^7-3 z^5 a^7+14 z^3 a^7-12 z a^7+2 a^7 z^{-1} +4 z^8 a^6-17 z^6 a^6+27 z^4 a^6-22 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5-z^7 a^5-6 z^5 a^5+10 z^3 a^5-8 z a^5+2 a^5 z^{-1} +2 z^8 a^4-6 z^6 a^4+3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+2 z^7 a^3-7 z^5 a^3+5 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               3 1   2 -5             3 2     -1 -7           3 2       1 -9       1 3 3         1 -11       5 3           2 -13     1 5             4 -15   1 2               -1 -17   1                 1 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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