L10a113

L10a112

L10a114

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a113 at Knotilus!

Link Presentations

[edit Notes on L10a113's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X20,8,11,7 X6,20,7,19 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code	{1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, {9, -1, 3, -7, 6, -8, 10, -3, 5, -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{(t(1)-1) \left(t(1)^2-t(1)+1\right) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 10 q^{9/2}-11 q^{7/2}+11 q^{5/2}-\frac{1}{q^{5/2}}-12 q^{3/2}+\frac{3}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-7 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +2 a z-7 z a^{-1} +8 z a^{-3} -3 z a^{-5} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^9 a^{-1} -2 z^9 a^{-3} -10 z^8 a^{-2} -7 z^8 a^{-4} -3 z^8-a z^7+z^7 a^{-1} -9 z^7 a^{-3} -11 z^7 a^{-5} +32 z^6 a^{-2} +10 z^6 a^{-4} -10 z^6 a^{-6} +12 z^6+4 a z^5+18 z^5 a^{-1} +44 z^5 a^{-3} +24 z^5 a^{-5} -6 z^5 a^{-7} -23 z^4 a^{-2} +10 z^4 a^{-4} +16 z^4 a^{-6} -3 z^4 a^{-8} -14 z^4-6 a z^3-30 z^3 a^{-1} -44 z^3 a^{-3} -16 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} -11 z^2 a^{-4} -8 z^2 a^{-6} +3 z^2+4 a z+14 z a^{-1} +16 z a^{-3} +6 z a^{-5} +1-a z^{-1} - a^{-1} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               6 2     -4 8             5 4       1 6           6 6         0 4         6 5           1 2       4 8             4 0     2 4               -2 -2   1 4                 3 -4   2                   -2 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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.

L10a112 L10a114