L11a25

L11a24

L11a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a25 at Knotilus!

Link Presentations

[edit Notes on L11a25's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 4, -8, 7, -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{u v^5-4 u v^4+6 u v^3-7 u v^2+6 u v-3 u-3 v^5+6 v^4-7 v^3+6 v^2-4 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 16 q^{9/2}-17 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-15 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} -2 z^5 a^{-5} -5 z^3 a^{-5} -2 z a^{-5} + a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} - a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-5 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -8 z^9 a^{-3} -5 z^9 a^{-5} -8 z^8 a^{-2} -15 z^8 a^{-4} -9 z^8 a^{-6} -2 z^8-a z^7+8 z^7 a^{-1} +16 z^7 a^{-3} -2 z^7 a^{-5} -9 z^7 a^{-7} +35 z^6 a^{-2} +49 z^6 a^{-4} +14 z^6 a^{-6} -6 z^6 a^{-8} +6 z^6+5 a z^5-4 z^5 a^{-1} -z^5 a^{-3} +25 z^5 a^{-5} +14 z^5 a^{-7} -3 z^5 a^{-9} -38 z^4 a^{-2} -49 z^4 a^{-4} -7 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -3 z^4-9 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -22 z^3 a^{-5} -12 z^3 a^{-7} +2 z^3 a^{-9} +14 z^2 a^{-2} +23 z^2 a^{-4} -4 z^2 a^{-8} +z^2 a^{-10} -4 z^2+7 a z+3 z a^{-1} -4 z a^{-3} +3 z a^{-5} +3 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             8 7         1 6           9 9           0 4         6 8             -2 2       5 10               5 0     2 5                 -3 -2     5                   5 -4 1 2                     -1 -6 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=4 }[/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}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

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