L11a344

L11a343

L11a345

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a344 at Knotilus!

Link Presentations

[edit Notes on L11a344's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,7,19,8 X10,5,1,6 X6374 X4,9,5,10 X20,14,21,13 X22,16,11,15 X14,22,15,21 X16,20,17,19 X2,11,3,12 X8,17,9,18
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}

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

L11a343 L11a345