L11a13

L11a12

L11a14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a13 at Knotilus!

Link Presentations

[edit Notes on L11a13's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X12,6,13,5 X8493 X16,10,17,9 X18,13,19,14 X14,17,15,18 X10,16,11,15 X22,12,5,11 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 10, -4, 7, -8, 9, -6, 8, -7, 11, -2, 3, -10}

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{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-9 q^{9/2}+16 q^{7/2}-22 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+4 a z^3-5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^3 z+3 a z-4 z a^{-1} +3 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10}-6 a z^9-13 z^9 a^{-1} -7 z^9 a^{-3} -7 a^2 z^8-19 z^8 a^{-2} -10 z^8 a^{-4} -16 z^8-4 a^3 z^7+2 a z^7+11 z^7 a^{-1} -3 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+14 a^2 z^6+43 z^6 a^{-2} +12 z^6 a^{-4} -4 z^6 a^{-6} +42 z^6+9 a^3 z^5+18 a z^5+20 z^5 a^{-1} +23 z^5 a^{-3} +11 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-8 a^2 z^4-27 z^4 a^{-2} -4 z^4 a^{-4} +5 z^4 a^{-6} -28 z^4-7 a^3 z^3-18 a z^3-23 z^3 a^{-1} -20 z^3 a^{-3} -7 z^3 a^{-5} +z^3 a^{-7} -a^4 z^2+a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+2 a^3 z+6 a z+8 z a^{-1} +6 z a^{-3} +2 z a^{-5} +1-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   6 1   5 8                 10 3     -7 6               12 6       6 4             12 10         -2 2           13 12           1 0         10 14             4 -2       6 11               -5 -4     3 10                 7 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

L11a12 L11a14