L11a521

L11a520

L11a522

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Link Presentations

[edit Notes on L11a521's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X10,14,11,13 X22,10,13,9 X20,17,21,18 X18,8,19,7 X12,20,7,19 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,1,21
Gauss code	{1, -9, 2, -10, 8, -11}, {6, -1, 4, -3, 9, -7}, {3, -2, 10, -8, 5, -6, 7, -5, 11, -4}

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^2 v^2 w^3-2 u^2 v^2 w^2-2 u^2 v w^3+4 u^2 v w^2-2 u^2 v w+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-2 u v^2 w+2 u v w^3-5 u v w^2+5 u v w-2 u v+2 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-3 q^9+7 q^8-11 q^7+16 q^6-17 q^5+18 q^4-15 q^3+12 q^2-7 q+4- q^{-1} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-8} +3 z^2 a^{-8} + a^{-8} z^{-2} +2 a^{-8} -2 z^6 a^{-6} -8 z^4 a^{-6} -10 z^2 a^{-6} -2 a^{-6} z^{-2} -6 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +9 z^4 a^{-4} +8 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} - a^{-10} +8 z^7 a^{-9} -10 z^5 a^{-9} +5 z^3 a^{-9} +9 z^8 a^{-8} -17 z^6 a^{-8} +17 z^4 a^{-8} -12 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +6 z^9 a^{-7} -5 z^7 a^{-7} -12 z^5 a^{-7} +17 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +10 z^8 a^{-6} -43 z^6 a^{-6} +50 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +11 z^9 a^{-5} -30 z^7 a^{-5} +16 z^5 a^{-5} +7 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +5 z^8 a^{-4} -35 z^6 a^{-4} +41 z^4 a^{-4} -17 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +5 z^9 a^{-3} -16 z^7 a^{-3} +12 z^5 a^{-3} -z^3 a^{-3} +4 z^8 a^{-2} -15 z^6 a^{-2} +16 z^4 a^{-2} -4 z^2 a^{-2} - a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +2 z^3 a^{-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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3 1 -2 17                   4     4 15                 7 3     -4 13               9 4       5 11             8 7         -1 9           10 9           1 7         7 10             3 5       5 8               -3 3     3 8                 5 1   1 4                   -3 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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