L11a87

L11a86

L11a88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a87 at Knotilus!

Link Presentations

[edit Notes on L11a87's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X10,14,11,13 X22,17,5,18 X18,7,19,8 X14,20,15,19 X16,10,17,9 X20,16,21,15 X8,21,9,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -9, 7, -3, 11, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -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 v^5-5 u v^4+10 u v^3-12 u v^2+7 u v-2 u-2 v^5+7 v^4-12 v^3+10 v^2-5 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{14}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+5 q^{5/2}-\frac{24}{q^{5/2}}-11 q^{3/2}+\frac{24}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{7}{q^{11/2}}+17 \sqrt{q}-\frac{22}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-9 a^3 z-4 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +4 a z^3-z^3 a^{-1} +2 a z+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-12 a^3 z^9-8 a z^9-4 a^6 z^8-9 a^4 z^8-18 a^2 z^8-13 z^8-3 a^7 z^7-a^5 z^7+12 a^3 z^7-a z^7-11 z^7 a^{-1} -a^8 z^6+4 a^6 z^6+19 a^4 z^6+37 a^2 z^6-5 z^6 a^{-2} +18 z^6+8 a^7 z^5+18 a^5 z^5+14 a^3 z^5+20 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+7 a^6 z^4-a^4 z^4-13 a^2 z^4+4 z^4 a^{-2} -4 z^4-8 a^7 z^3-25 a^5 z^3-25 a^3 z^3-12 a z^3-4 z^3 a^{-1} -3 a^8 z^2-11 a^6 z^2-15 a^4 z^2-8 a^2 z^2-z^2+4 a^7 z+16 a^5 z+15 a^3 z+3 a z+a^8+4 a^6+7 a^4+4 a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -4 a^3 z^{-1} -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                   7 1   6 2                 10 4     -6 0               12 7       5 -2             13 11         -2 -4           11 11           0 -6         8 13             5 -8       6 11               -5 -10     2 9                 7 -12   1 5                   -4 -14   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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.

L11a86 L11a88