L11a323

L11a322

L11a324

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a323 at Knotilus!

Link Presentations

[edit Notes on L11a323's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X14,7,15,8 X12,4,13,3 X16,14,17,13 X6,15,7,16 X2,18,3,17
Gauss code	{1, -11, 8, -6, 5, -10, 7, -3}, {3, -1, 2, -8, 9, -7, 10, -9, 11, -2, 4, -5, 6, -4}

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{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-3 u v^2+5 u v-3 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+21 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z-2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -3 a^3 z+z a^{-3} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-4 z^3 a^{-1} +4 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^{10}-2 z^{10}-7 a^3 z^9-13 a z^9-6 z^9 a^{-1} -10 a^4 z^8-19 a^2 z^8-7 z^8 a^{-2} -16 z^8-8 a^5 z^7-3 a^3 z^7+11 a z^7+2 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+12 a^4 z^6+43 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +42 z^6-a^7 z^5+11 a^5 z^5+23 a^3 z^5+20 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-27 a^2 z^4-8 z^4 a^{-2} +2 z^4 a^{-4} -28 z^4+a^7 z^3-7 a^5 z^3-20 a^3 z^3-23 a z^3-18 z^3 a^{-1} -7 z^3 a^{-3} -2 a^6 z^2-a^4 z^2+5 a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +6 z^2+2 a^5 z+6 a^3 z+8 a z+6 z a^{-1} +2 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 10 3     7 2               11 6       -5 0             14 10         4 -2           12 13           1 -4         10 12             -2 -6       6 12               6 -8     3 10                 -7 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L11a322 L11a324