L11n365

L11n364

L11n366

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Link Presentations

[edit Notes on L11n365's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X9,18,10,19 X17,8,18,9 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -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{(u-1) (w-1) \left(-v^2 w+v w^3-v w^2+v w-v+w^2\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -1+4 q^{-1} -5 q^{-2} +8 q^{-3} -8 q^{-4} +8 q^{-5} -6 q^{-6} +5 q^{-7} -2 q^{-8} + q^{-9} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^4-6 a^6 z^2-2 a^6 z^{-2} -6 a^6+a^4 z^6+4 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^4-a^2 z^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-6 z^5 a^9+3 z^3 a^9+z a^9+2 z^8 a^8-4 z^6 a^8-z^4 a^8-z^2 a^8-a^8 z^{-2} +3 a^8+z^9 a^7+z^7 a^7-9 z^5 a^7+9 z^3 a^7-7 z a^7+2 a^7 z^{-1} +5 z^8 a^6-17 z^6 a^6+25 z^4 a^6-23 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5+z^7 a^5-8 z^5 a^5+14 z^3 a^5-9 z a^5+2 a^5 z^{-1} +3 z^8 a^4-12 z^6 a^4+26 z^4 a^4-21 z^2 a^4-a^4 z^{-2} +7 a^4+2 z^7 a^3-5 z^5 a^3+9 z^3 a^3-2 z a^3+4 z^4 a^2-4 z^2 a^2+z^3 a-z a }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                   1 -1 -1                 3   3 -3               4 3   -1 -5             4 1     3 -7           4 4       0 -9         4 4         0 -11       2 4           2 -13     3 4             -1 -15   1 4               3 -17   1                 -1 -19 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=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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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