L11a14

L11a13

L11a15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a14 at Knotilus!

Link Presentations

[edit Notes on L11a14's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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(2 v^4-4 v^3+5 v^2-4 v+2\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 22 q^{9/2}-22 q^{7/2}+17 q^{5/2}-13 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+5 q^{17/2}-10 q^{15/2}+16 q^{13/2}-20 q^{11/2}+6 \sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-7} -z^3 a^{-7} +2 z a^{-7} +2 a^{-7} z^{-1} +z^7 a^{-5} +2 z^5 a^{-5} -3 z^3 a^{-5} -9 z a^{-5} -5 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +9 z a^{-3} +3 a^{-3} z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} +5 z^6 a^{-10} -5 z^4 a^{-10} + a^{-10} +10 z^7 a^{-9} -14 z^5 a^{-9} +3 z^3 a^{-9} +11 z^8 a^{-8} -14 z^6 a^{-8} +z^4 a^{-8} +7 z^9 a^{-7} -z^7 a^{-7} -12 z^5 a^{-7} +6 z^3 a^{-7} -3 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +13 z^8 a^{-6} -28 z^6 a^{-6} +12 z^4 a^{-6} +6 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -17 z^7 a^{-5} +2 z^5 a^{-5} +14 z^3 a^{-5} -13 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +6 z^8 a^{-4} -17 z^6 a^{-4} +9 z^4 a^{-4} +7 z^2 a^{-4} -5 a^{-4} +4 z^9 a^{-3} -3 z^7 a^{-3} -10 z^5 a^{-3} +20 z^3 a^{-3} -14 z a^{-3} +3 a^{-3} z^{-1} +4 z^8 a^{-2} -7 z^6 a^{-2} +3 z^2 a^{-2} +3 z^7 a^{-1} -9 z^5 a^{-1} +9 z^3 a^{-1} -4 z a^{-1} +z^6-3 z^4+2 z^2 }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     4   -4 16                   6 1   5 14                 10 4     -6 12               10 6       4 10             12 10         -2 8           10 10           0 6         7 12             5 4       6 10               -4 2     2 9                 7 0   1 4                   -3 -2   2                     2 -4 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/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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