L11a459

L11a458

L11a460

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a459 at Knotilus!

Link Presentations

[edit Notes on L11a459's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X22,20,13,19 X18,8,19,7 X10,14,11,13 X8,21,9,22 X20,9,21,10 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}

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(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+5 t(1) t(3)^2 t(2)^2-5 t(3)^2 t(2)^2+t(1) t(2)^2-4 t(1) t(3) t(2)^2+5 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-t(3)^3 t(2)-5 t(1) t(3)^2 t(2)+4 t(3)^2 t(2)-2 t(1) t(2)+5 t(1) t(3) t(2)-5 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-4 q^{-5} +4 q^4+9 q^{-4} -8 q^3-13 q^{-3} +13 q^2+19 q^{-2} -18 q-20 q^{-1} +22 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^6-z^6 a^{-2} -7 a^2 z^4-3 z^4 a^{-2} -7 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -3 a^2+z^8+5 z^6+9 z^4+6 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+12 a z^9+6 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+8 z^8 a^{-2} +12 z^8+4 a^5 z^7-9 a^3 z^7-24 a z^7-4 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-39 a^2 z^6-12 z^6 a^{-2} +4 z^6 a^{-4} -37 z^6-9 a^5 z^5-2 a^3 z^5+14 a z^5-5 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+40 a^2 z^4+7 z^4 a^{-2} -6 z^4 a^{-4} +39 z^4+4 a^5 z^3+4 a^3 z^3+a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-7 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} -14 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 8 3     5 3               10 5       -5 1             12 8         4 -1           11 13           2 -3         8 9             -1 -5       5 11               6 -7     4 8                 -4 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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.

L11a458 L11a460