L11a317

L11a316

L11a318

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a317 at Knotilus!

Link Presentations

[edit Notes on L11a317's Link Presentations]

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

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{(u-1) (v-1) \left(u^2 v^3-2 u^2 v^2+u^2 v+u v^4-3 u v^3+6 u v^2-3 u v+u+v^3-2 v^2+v\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{17}{q^{9/2}}-q^{7/2}+\frac{24}{q^{7/2}}+5 q^{5/2}-\frac{29}{q^{5/2}}-12 q^{3/2}+\frac{28}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{10}{q^{11/2}}+19 \sqrt{q}-\frac{26}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+2 a z^5-z^5 a^{-1} -2 a^5 z^3+5 a^3 z^3-z^3 a^{-1} -2 a^5 z+5 a^3 z-3 a z-a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -4 a^4 z^{10}-4 a^2 z^{10}-9 a^5 z^9-21 a^3 z^9-12 a z^9-8 a^6 z^8-11 a^4 z^8-19 a^2 z^8-16 z^8-4 a^7 z^7+16 a^5 z^7+40 a^3 z^7+8 a z^7-12 z^7 a^{-1} -a^8 z^6+17 a^6 z^6+38 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5-11 a^5 z^5-26 a^3 z^5+9 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-13 a^6 z^4-32 a^4 z^4-29 a^2 z^4+3 z^4 a^{-2} -9 z^4-4 a^7 z^3+7 a^5 z^3+15 a^3 z^3-a z^3-5 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+12 a^4 z^2+8 a^2 z^2+2 z^2-4 a^5 z-9 a^3 z-5 a z-a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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                     4   -4 4                   8 1   7 2                 11 4     -7 0               15 8       7 -2             15 13         -2 -4           14 13           1 -6         10 15             5 -8       7 14               -7 -10     3 10                 7 -12   1 7                   -6 -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}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

L11a316 L11a318