L10a16

L10a15

L10a17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a16 at Knotilus!

Link Presentations

[edit Notes on L10a16's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X20,11,5,12 X18,13,19,14 X14,17,15,18 X12,19,13,20 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 5, -6, 8, -3, 6, -5, 7, -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{2 t(2)^3+3 t(1) t(2)^2-6 t(2)^2-6 t(1) t(2)+3 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{7}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -3 a^7 z-2 a^7 z^{-1} +2 a^5 z^3+2 a^5 z+a^5 z^{-1} +2 a^3 z^3+a^3 z+a^3 z^{-1} +a z^3-a z-a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+9 z^6 a^8-11 z^4 a^8+4 z^2 a^8-a^8-z^9 a^7+15 z^5 a^7-26 z^3 a^7+13 z a^7-2 a^7 z^{-1} -5 z^8 a^6+20 z^6 a^6-23 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5-2 z^7 a^5+16 z^5 a^5-18 z^3 a^5+8 z a^5-a^5 z^{-1} -3 z^8 a^4+8 z^6 a^4-8 z^4 a^4+7 z^2 a^4-2 a^4-3 z^7 a^3+3 z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -3 z^6 a^2+2 z^4 a^2+z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                     1 1 2                   1   -1 0                 3 1   2 -2               3 2     -1 -4             4 2       2 -6           3 3         0 -8         3 4           -1 -10       3 4             1 -12     1 2               -1 -14   1 3                 2 -16   1                   -1 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10a15 L10a17