L11a540

L11a539

L11a541

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a540 at Knotilus!

Link Presentations

[edit Notes on L11a540's Link Presentations]

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

A Braid Representative
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{(x-1) \left(u v w x^2-2 u v w x+2 u v x-u v-u w x^2+3 u w x-2 u x+2 u-2 v w x^2+2 v w x-3 v x+v+w x^2-2 w x+2 x-1\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-3} +a^9 z+2 a^9 z^{-1} -3 a^7 z^3-3 a^7 z^{-3} -9 a^7 z-9 a^7 z^{-1} +3 a^5 z^5+11 a^5 z^3+3 a^5 z^{-3} +17 a^5 z+12 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-8 a^3 z^3-a^3 z^{-3} -10 a^3 z-5 a^3 z^{-1} +a z^5+2 a z^3+a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+3 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -2 z^6 a^{10}+3 z^4 a^{10}-a^{10}-3 z^7 a^9+3 z^5 a^9-z^3 a^9+3 z a^9-3 a^9 z^{-1} +a^9 z^{-3} -3 z^8 a^8-2 z^6 a^8+12 z^4 a^8-16 z^2 a^8-3 a^8 z^{-2} +11 a^8-3 z^9 a^7-z^7 a^7+9 z^5 a^7-20 z^3 a^7+20 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -z^{10} a^6-9 z^8 a^6+19 z^6 a^6-3 z^4 a^6-28 z^2 a^6-6 a^6 z^{-2} +24 a^6-7 z^9 a^5+4 z^7 a^5+23 z^5 a^5-39 z^3 a^5+29 z a^5-14 a^5 z^{-1} +3 a^5 z^{-3} -z^{10} a^4-12 z^8 a^4+32 z^6 a^4-16 z^4 a^4-12 z^2 a^4-3 a^4 z^{-2} +13 a^4-4 z^9 a^3-2 z^7 a^3+28 z^5 a^3-31 z^3 a^3+18 z a^3-6 a^3 z^{-1} +a^3 z^{-3} -6 z^8 a^2+12 z^6 a^2-2 z^4 a^2-z^2 a^2-4 z^7 a+10 z^5 a-8 z^3 a+3 z a-z^6+2 z^4-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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               9 8       -1 -6             10 6         4 -8           7 9           2 -10         9 10             -1 -12       4 11               7 -14     2 5                 -3 -16     4                   4 -18 1 2                     -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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.

L11a539 L11a541