L11a316

L11a315

L11a317

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a316 at Knotilus!

Link Presentations

[edit Notes on L11a316's Link Presentations]

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

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^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+3 u^2 v-u^2-u v^5+3 u v^4-5 u v^3+5 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{14}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{15}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+6 a^7 z^3+6 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-4 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+5 a^{12} z^6-5 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3-2 a^{11} z+8 a^{10} z^8-18 a^{10} z^6+18 a^{10} z^4-6 a^{10} z^2+a^{10}+6 a^9 z^9-12 a^9 z^7+6 a^9 z^5+a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-33 a^8 z^6+39 a^8 z^4-18 a^8 z^2+3 a^8+10 a^7 z^9-33 a^7 z^7+35 a^7 z^5-22 a^7 z^3+11 a^7 z-3 a^7 z^{-1} +2 a^6 z^{10}+a^6 z^8-21 a^6 z^6+26 a^6 z^4-13 a^6 z^2+3 a^6+4 a^5 z^9-13 a^5 z^7+11 a^5 z^5-7 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+11 a^4 z^4-3 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-2 a^3 z }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     2   -2 -4                   5 1   4 -6                 6 3     -3 -8               8 4       4 -10             7 6         -1 -12           9 8           1 -14         6 8             2 -16       4 8               -4 -18     2 6                 4 -20   1 4                   -3 -22   2                     2 -24 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a315 L11a317