L10a2

L10a1

L10a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a2 at Knotilus!

Link Presentations

[edit Notes on L10a2's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X20,13,5,14 X18,11,19,12 X12,19,13,20 X2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, 9, -7}

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{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{14}{q^{9/2}}-\frac{15}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+5 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3+3 a^3 z^{-1} +a z^5+2 a z^3-a z-2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4+4 a^9 z^5-2 a^9 z^3+8 a^8 z^6-9 a^8 z^4+3 a^8 z^2+9 a^7 z^7-11 a^7 z^5+5 a^7 z^3-2 a^7 z+6 a^6 z^8-a^6 z^6-9 a^6 z^4+2 a^6 z^2+a^6+2 a^5 z^9+11 a^5 z^7-30 a^5 z^5+20 a^5 z^3-4 a^5 z-a^5 z^{-1} +11 a^4 z^8-23 a^4 z^6+12 a^4 z^4-4 a^4 z^2+3 a^4+2 a^3 z^9+6 a^3 z^7-27 a^3 z^5+22 a^3 z^3-a^3 z-3 a^3 z^{-1} +5 a^2 z^8-13 a^2 z^6+9 a^2 z^4-3 a^2 z^2+3 a^2+4 a z^7-12 a z^5+9 a z^3+a z-2 a z^{-1} +z^6-2 z^4 }[/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 χ 4                     1 -1 2                   3   3 0                 3 1   -2 -2               8 3     5 -4             7 5       -2 -6           8 6         2 -8         6 7           1 -10       5 8             -3 -12     3 6               3 -14   1 5                 -4 -16   3                   3 -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/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}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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.

L10a1 L10a3