L11a311

L11a310

L11a312

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a311 at Knotilus!

Link Presentations

[edit Notes on L11a311's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,18,21,17 X18,5,19,6 X4,19,5,20 X14,7,15,8 X16,13,17,14 X8,15,1,16 X6,22,7,21
Gauss code	{1, -4, 2, -7, 6, -11, 8, -10}, {4, -1, 3, -2, 9, -8, 10, -9, 5, -6, 7, -5, 11, -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{u^3 v^3-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+10 u^2 v^2-11 u^2 v+3 u^2+3 u v^3-11 u v^2+10 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{14}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+4 q^{5/2}-\frac{23}{q^{5/2}}-9 q^{3/2}+\frac{22}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+3 a^5 z^3+4 a^5 z+a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-8 a^3 z-a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} +5 a z-2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-9 a^3 z^9-5 a z^9-6 a^6 z^8-17 a^4 z^8-20 a^2 z^8-9 z^8-4 a^7 z^7-5 a^5 z^7-3 a^3 z^7-10 a z^7-8 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+41 a^4 z^6+42 a^2 z^6-4 z^6 a^{-2} +10 z^6+10 a^7 z^5+30 a^5 z^5+42 a^3 z^5+35 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-6 a^6 z^4-27 a^4 z^4-26 a^2 z^4+5 z^4 a^{-2} -2 z^4-8 a^7 z^3-29 a^5 z^3-41 a^3 z^3-29 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+4 a^4 z^2+5 a^2 z^2-2 z^2 a^{-2} +2 a^7 z+10 a^5 z+13 a^3 z+8 a z+3 z a^{-1} +a^4-a^5 z^{-1} -a^3 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               11 6       5 -2             12 10         -2 -4           11 10           1 -6         8 12             4 -8       6 11               -5 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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.

L11a310 L11a312