L10a142

L10a141

L10a143

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a142 at Knotilus!

Link Presentations

[edit Notes on L10a142's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X12,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X20,11,13,12 X10,19,11,20 X2536 X4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -4, 6, -8, 7, -3}, {10, -2, 3, -5, 4, -6, 8, -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{u v^3 w^2+u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^2-2 u v w+u v+u w-u+v^3 w^3-v^3 w^2-v^2 w^3+2 v^2 w^2-v^2 w-v w^2+2 v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} -2 q^{-4} +4 q^{-5} -4 q^{-6} +7 q^{-7} -7 q^{-8} +7 q^{-9} -5 q^{-10} +4 q^{-11} -2 q^{-12} + q^{-13} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^2+a^{12} z^{-2} +3 a^{12}-3 a^{10} z^4-12 a^{10} z^2-2 a^{10} z^{-2} -11 a^{10}+2 a^8 z^6+10 a^8 z^4+15 a^8 z^2+a^8 z^{-2} +8 a^8+a^6 z^6+4 a^6 z^4+3 a^6 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-3 a^{15} z^3+2 a^{14} z^6-a^{14} z^4-2 a^{14} z^2+2 a^{13} z^7-2 a^{13} z^5+a^{13} z^3+2 a^{12} z^8-5 a^{12} z^6+9 a^{12} z^4-8 a^{12} z^2-a^{12} z^{-2} +5 a^{12}+a^{11} z^9-8 a^{11} z^5+18 a^{11} z^3-11 a^{11} z+2 a^{11} z^{-1} +5 a^{10} z^8-22 a^{10} z^6+38 a^{10} z^4-31 a^{10} z^2-2 a^{10} z^{-2} +13 a^{10}+a^9 z^9-11 a^9 z^5+18 a^9 z^3-11 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-14 a^8 z^6+23 a^8 z^4-20 a^8 z^2-a^8 z^{-2} +8 a^8+2 a^7 z^7-7 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+3 a^6 z^2 }[/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 -5                     1 1 -7                   2 1 -1 -9                 2     2 -11               2 2     0 -13             5 2       3 -15           3 3         0 -17         4 4           0 -19       1 3             2 -21     3 4               -1 -23   1 3                 2 -25   1                   -1 -27 1                     1

Integral Khovanov Homology (db, data source)

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

L10a141 L10a143