L11a60

L11a59

L11a61

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a60 at Knotilus!

Link Presentations

[edit Notes on L11a60's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X20,18,21,17 X18,9,19,10 X8,19,9,20 X22,16,5,15 X16,22,17,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 7, -8, 4, -5, 6, -4, 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{2 t(1) t(2)^3-4 t(2)^3-8 t(1) t(2)^2+11 t(2)^2+11 t(1) t(2)-8 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{6}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+7 q^{5/2}-\frac{13}{q^{5/2}}-11 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+14 \sqrt{q}-\frac{16}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3+z^3 a^{-3} +a^3 z-a^3 z^{-1} -a z^5-z^5 a^{-1} +a z^3-z^3 a^{-1} +2 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-2 a^3 z^9-6 a z^9-4 z^9 a^{-1} -3 a^4 z^8-5 a^2 z^8-6 z^8 a^{-2} -8 z^8-3 a^5 z^7-4 a^3 z^7+8 a z^7+5 z^7 a^{-1} -4 z^7 a^{-3} -2 a^6 z^6+a^4 z^6+10 a^2 z^6+17 z^6 a^{-2} -z^6 a^{-4} +25 z^6-a^7 z^5+3 a^5 z^5+12 a^3 z^5+5 a z^5+8 z^5 a^{-1} +11 z^5 a^{-3} +3 a^6 z^4+a^4 z^4-9 a^2 z^4-12 z^4 a^{-2} +2 z^4 a^{-4} -21 z^4+3 a^7 z^3+a^5 z^3-14 a^3 z^3-18 a z^3-12 z^3 a^{-1} -6 z^3 a^{-3} +2 a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} +6 z^2-3 a^7 z-2 a^5 z+8 a^3 z+12 a z+5 z a^{-1} -a^6-2 a^4-3 a^2-1+a^7 z^{-1} +a^5 z^{-1} -a^3 z^{-1} -2 a z^{-1} - a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   4 1   -3 4                 7 3     4 2               7 4       -3 0             9 7         2 -2           8 8           0 -4         5 8             -3 -6       4 8               4 -8     2 5                 -3 -10   1 5                   4 -12   1                     -1 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=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.

L11a59 L11a61