L11a481

L11a480

L11a482

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a481 at Knotilus!

Link Presentations

[edit Notes on L11a481's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X20,16,21,15 X18,10,19,9 X8,18,9,17 X22,20,17,19 X16,22,5,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8}

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{(w-1) \left(2 u v w^2-4 u v w+2 u v-u w^2+3 u w-4 u-4 v w^2+3 v w-v+2 w^2-4 w+2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+5 q^7-10 q^6+15 q^5-19 q^4+21 q^3-20 q^2+17 q-10+7 q^{-1} -2 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-z^2 a^{-4} -4 z^2+2 a^2+ a^{-2} - a^{-4} + a^{-6} -3+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} +5 z^6 a^{-8} -5 z^4 a^{-8} +10 z^7 a^{-7} -15 z^5 a^{-7} +4 z^3 a^{-7} +10 z^8 a^{-6} -11 z^6 a^{-6} -4 z^4 a^{-6} +6 z^2 a^{-6} -2 a^{-6} +5 z^9 a^{-5} +8 z^7 a^{-5} -28 z^5 a^{-5} +18 z^3 a^{-5} -4 z a^{-5} +z^{10} a^{-4} +15 z^8 a^{-4} -24 z^6 a^{-4} -z^4 a^{-4} +12 z^2 a^{-4} -2 a^{-4} +7 z^9 a^{-3} +2 z^7 a^{-3} -30 z^5 a^{-3} +34 z^3 a^{-3} -12 z a^{-3} +z^{10} a^{-2} +8 z^8 a^{-2} +a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4+6 a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-2 a^{-2} +2 z^9 a^{-1} +2 a z^7+6 z^7 a^{-1} -4 a z^5-22 z^5 a^{-1} +20 z^3 a^{-1} +4 a z-4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-4 z^6-2 z^4+6 z^2+2 z^{-2} -5 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     4   4 13                   6 1   -5 11                 9 4     5 9               11 7       -4 7             10 8         2 5           10 11           1 3         7 10             -3 1       5 12               7 -1     2 5                 -3 -3     5                   5 -5 1 2                     -1 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/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.

L11a480 L11a482