L10a101

L10a100

L10a102

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a101 at Knotilus!
	Contains two L4a1 configurations.

As a Celtic (or pseudo-Celtic) linear decorative knot

(multicolored)

Mongolian ornament

Rotated knotwork cross with four L10a101 sub-configurations

Link Presentations

[edit Notes on L10a101's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,19,15,20 X18,7,19,8 X6,15,7,16 X16,5,17,6 X4,17,5,18 X20,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code	{1, -9, 2, -7, 6, -5, 4, -10}, {9, -1, 10, -2, 8, -3, 5, -6, 7, -4, 3, -8}

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)+t(2)) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{11}-z^3 a^9+z a^9+a^9 z^{-1} -3 z^3 a^7-3 z a^7-a^7 z^{-1} -3 z^3 a^5-3 z a^5-z^3 a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^7-5 a^{13} z^5+8 a^{13} z^3-4 a^{13} z+2 a^{12} z^8-9 a^{12} z^6+12 a^{12} z^4-4 a^{12} z^2+a^{11} z^9-11 a^{11} z^5+15 a^{11} z^3-3 a^{11} z+5 a^{10} z^8-15 a^{10} z^6+9 a^{10} z^4+a^{10} z^2+a^9 z^9+5 a^9 z^7-21 a^9 z^5+18 a^9 z^3-7 a^9 z+a^9 z^{-1} +3 a^8 z^8+a^8 z^6-15 a^8 z^4+8 a^8 z^2-a^8+6 a^7 z^7-9 a^7 z^5+4 a^7 z^3-5 a^7 z+a^7 z^{-1} +7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 z^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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                     1 1 -4                   3 1 -2 -6                 3     3 -8               4 3     -1 -10             5 3       2 -12           4 4         0 -14         4 5           -1 -16       2 4             2 -18     2 4               -2 -20   1 3                 2 -22   1                   -1 -24 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=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10a100 L10a102