L11a111

L11a110

L11a112

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Link Presentations

[edit Notes on L11a111's Link Presentations]

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

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(2 v^4-v^3+2 v^2-v+2\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{23/2}-2 q^{21/2}+4 q^{19/2}-6 q^{17/2}+9 q^{15/2}-10 q^{13/2}+9 q^{11/2}-9 q^{9/2}+6 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-9} +4 z^3 a^{-9} +4 z a^{-9} +2 a^{-9} z^{-1} -z^7 a^{-7} -5 z^5 a^{-7} -9 z^3 a^{-7} -9 z a^{-7} -4 a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-8} -3 z^8 a^{-10} -z^7 a^{-3} +6 z^7 a^{-5} +21 z^7 a^{-7} +11 z^7 a^{-9} -3 z^7 a^{-11} +8 z^6 a^{-4} +11 z^6 a^{-6} +12 z^6 a^{-8} +6 z^6 a^{-10} -3 z^6 a^{-12} +5 z^5 a^{-3} -z^5 a^{-5} -35 z^5 a^{-7} -22 z^5 a^{-9} +5 z^5 a^{-11} -2 z^5 a^{-13} -7 z^4 a^{-4} -18 z^4 a^{-6} -22 z^4 a^{-8} -4 z^4 a^{-10} +6 z^4 a^{-12} -z^4 a^{-14} -8 z^3 a^{-3} -2 z^3 a^{-5} +35 z^3 a^{-7} +22 z^3 a^{-9} -4 z^3 a^{-11} +3 z^3 a^{-13} -z^2 a^{-4} +15 z^2 a^{-6} +20 z^2 a^{-8} -4 z^2 a^{-10} -6 z^2 a^{-12} +2 z^2 a^{-14} +5 z a^{-3} -3 z a^{-5} -17 z a^{-7} -10 z a^{-9} -z a^{-11} + a^{-4} -5 a^{-6} -6 a^{-8} + a^{-10} +2 a^{-12} - a^{-3} z^{-1} + a^{-5} z^{-1} +4 a^{-7} z^{-1} +2 a^{-9} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     1   1 20                   3 1   -2 18                 3 1     2 16               6 3       -3 14             4 3         1 12           5 6           1 10         4 4             0 8       2 5               3 6     3 4                 -1 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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