L11a496

L11a495

L11a497

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Link Presentations

[edit Notes on L11a496's Link Presentations]

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

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{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(3)^4-t(3)^3+t(3)^2-t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-6 q^7+8 q^6-11 q^5+13 q^4-11 q^3+11 q^2-7 q+6-2 q^{-1} + q^{-2} }[/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} -10 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-17 z^2 a^{-2} +18 z^2 a^{-4} -5 z^2 a^{-6} +4 z^2-14 a^{-2} +13 a^{-4} -4 a^{-6} +5-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +3 z^8 a^{-2} +11 z^8 a^{-4} +9 z^8 a^{-6} +z^8-9 z^7 a^{-1} -25 z^7 a^{-3} -7 z^7 a^{-5} +9 z^7 a^{-7} -34 z^6 a^{-2} -60 z^6 a^{-4} -24 z^6 a^{-6} +8 z^6 a^{-8} -6 z^6+10 z^5 a^{-1} +10 z^5 a^{-3} -22 z^5 a^{-5} -16 z^5 a^{-7} +6 z^5 a^{-9} +70 z^4 a^{-2} +88 z^4 a^{-4} +19 z^4 a^{-6} -10 z^4 a^{-8} +3 z^4 a^{-10} +14 z^4+4 z^3 a^{-1} +34 z^3 a^{-3} +41 z^3 a^{-5} +5 z^3 a^{-7} -5 z^3 a^{-9} +z^3 a^{-11} -57 z^2 a^{-2} -57 z^2 a^{-4} -14 z^2 a^{-6} +2 z^2 a^{-8} -16 z^2-12 z a^{-1} -31 z a^{-3} -25 z a^{-5} -4 z a^{-7} +2 z a^{-9} +23 a^{-2} +22 a^{-4} +7 a^{-6} +9+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 4 2     2 11               7 4       -3 9             6 4         2 7           5 7           2 5         6 6             0 3       5 9               4 1     1 2                 -1 -1   1 5                   4 -3   1                     -1 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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