L11a393

L11a392

L11a394

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Link Presentations

[edit Notes on L11a393's Link Presentations]

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

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 w^5-u v w^4+u v w^3-u v w^2+u v w-u v-u w^5+2 u w^4-2 u w^3+2 u w^2-2 u w+2 u-2 v w^5+2 v w^4-2 v w^3+2 v w^2-2 v w+v+w^5-w^4+w^3-w^2+w-1}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-6 q^7+7 q^6-10 q^5+11 q^4-9 q^3+9 q^2+ q^{-2} -5 q- q^{-1} +5 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -11 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-21 z^2 a^{-2} +20 z^2 a^{-4} -5 z^2 a^{-6} +5 z^2-20 a^{-2} +18 a^{-4} -5 a^{-6} +7-8 a^{-2} z^{-2} +7 a^{-4} z^{-2} -2 a^{-6} z^{-2} +3 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +5 z^9 a^{-3} +4 z^9 a^{-5} -z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +z^8-3 z^7 a^{-1} -18 z^7 a^{-3} -8 z^7 a^{-5} +7 z^7 a^{-7} -15 z^6 a^{-2} -34 z^6 a^{-4} -19 z^6 a^{-6} +7 z^6 a^{-8} -7 z^6-5 z^5 a^{-1} +2 z^5 a^{-3} -9 z^5 a^{-5} -10 z^5 a^{-7} +6 z^5 a^{-9} +42 z^4 a^{-2} +51 z^4 a^{-4} +16 z^4 a^{-6} -8 z^4 a^{-8} +3 z^4 a^{-10} +18 z^4+23 z^3 a^{-1} +44 z^3 a^{-3} +25 z^3 a^{-5} -3 z^3 a^{-7} -6 z^3 a^{-9} +z^3 a^{-11} -47 z^2 a^{-2} -37 z^2 a^{-4} -12 z^2 a^{-6} -22 z^2-24 z a^{-1} -45 z a^{-3} -21 z a^{-5} +3 z a^{-7} +3 z a^{-9} +28 a^{-2} +22 a^{-4} +7 a^{-6} + a^{-8} +13+8 a^{-1} z^{-1} +15 a^{-3} z^{-1} +7 a^{-5} z^{-1} - a^{-7} z^{-1} - a^{-9} z^{-1} -8 a^{-2} z^{-2} -7 a^{-4} z^{-2} -2 a^{-6} z^{-2} -3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 3 2     1 11               7 4       -3 9             4 3         1 7           5 7           2 5         4 4             0 3       4 8               4 1     1 1                 0 -1     4                   4 -3 1 1                     0 -5 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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