L11a372

L11a371

L11a373

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a372 at Knotilus!

Link Presentations

[edit Notes on L11a372's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X10,11,1,12 X4,15,5,16 X22,17,11,18 X18,10,19,9 X20,8,21,7 X8,20,9,19 X6,22,7,21
Gauss code	{1, -2, 3, -6, 4, -11, 9, -10, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -8, 10, -9, 11, -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)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-t(2)^3 t(1)^3+t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+t(2)^3 t(1)^2-t(2)^2 t(1)^2+t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+t(2)^2 t(1)-t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{5/2}}-q^{3/2}+\frac{3}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^7-4 z^3 a^7-3 z a^7+z^7 a^5+5 z^5 a^5+6 z^3 a^5+z a^5+z^7 a^3+6 z^5 a^3+11 z^3 a^3+7 z a^3+a^3 z^{-1} -z^5 a-5 z^3 a-6 z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{12}-2 z^3 a^{11}-3 z^4 a^{10}+2 z^2 a^{10}-4 z^5 a^9+7 z^3 a^9-2 z a^9-4 z^6 a^8+9 z^4 a^8-2 z^2 a^8-4 z^7 a^7+13 z^5 a^7-9 z^3 a^7+2 z a^7-3 z^8 a^6+11 z^6 a^6-8 z^4 a^6-2 z^9 a^5+8 z^7 a^5-5 z^5 a^5-5 z^3 a^5+2 z a^5-z^{10} a^4+4 z^8 a^4-z^6 a^4-7 z^4 a^4+3 z^2 a^4-3 z^9 a^3+20 z^7 a^3-44 z^5 a^3+37 z^3 a^3-11 z a^3+a^3 z^{-1} -z^{10} a^2+7 z^8 a^2-16 z^6 a^2+13 z^4 a^2-2 z^2 a^2-a^2-z^9 a+8 z^7 a-22 z^5 a+24 z^3 a-9 z a+a z^{-1} }[/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 4                       1 1 2                         0 0                   2 1   1 -2                 1       -1 -4               3 2       1 -6             2 2         0 -8           3 2           1 -10         2 3             1 -12       2 2               0 -14     1 2                 1 -16   1 2                   -1 -18   1                     1 -20 1                       -1

Integral Khovanov Homology (db, data source)

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

L11a371 L11a373