L10a60

L10a59

L10a61

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a60 at Knotilus!

Link Presentations

[edit Notes on L10a60's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X14,6,15,5 X18,13,19,14 X16,19,17,20 X12,17,13,18 X2738 X4,12,5,11 X6,16,1,15
Gauss code	{1, -8, 2, -9, 4, -10}, {8, -1, 3, -2, 9, -7, 5, -4, 10, -6, 7, -5, 6, -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^2 v^4-3 u^2 v^3+4 u^2 v^2-2 u^2 v-u v^4+4 u v^3-7 u v^2+4 u v-u-2 v^3+4 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-3 q^{9/2}+6 q^{7/2}-10 q^{5/2}+11 q^{3/2}-13 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -2 a^3 z+3 z a^{-3} -a^3 z^{-1} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +7 a z^3-10 z^3 a^{-1} +8 a z-8 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +a^4 z^6+5 z^6 a^{-4} -3 a^4 z^4-5 z^4 a^{-4} +3 a^4 z^2+2 z^2 a^{-4} -a^4+3 a^3 z^7+6 z^7 a^{-3} -9 a^3 z^5-8 z^5 a^{-3} +8 a^3 z^3+7 z^3 a^{-3} -3 a^3 z-3 z a^{-3} +a^3 z^{-1} +3 a^2 z^8+4 z^8 a^{-2} -4 a^2 z^6-7 a^2 z^4-8 z^4 a^{-2} +10 a^2 z^2+6 z^2 a^{-2} -3 a^2+a z^9+z^9 a^{-1} +8 a z^7+11 z^7 a^{-1} -29 a z^5-31 z^5 a^{-1} +28 a z^3+30 z^3 a^{-1} -13 a z-14 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +7 z^8-10 z^6-6 z^4+10 z^2-3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 4 1   -3 6               6 2     4 4             6 5       -1 2           7 5         2 0         5 7           2 -2       4 6             -2 -4     2 5               3 -6   1 4                 -3 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

L10a59 L10a61