L11n361

L11n360

L11n362

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Link Presentations

[edit Notes on L11n361's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X20,9,21,10 X10,19,11,20 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, 7, -6, -10, 9}, {-4, -1, 2, -5, 6, -7, 11, -2, 3, 8, -9, 10}

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)^2 \left(v^2+v w+w^2\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +3 q^{-9} -4 q^{-8} +7 q^{-7} -7 q^{-6} +8 q^{-5} -7 q^{-4} +6 q^{-3} -3 q^{-2} +2 q^{-1} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+3 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^4-5 a^6 z^2-2 a^6 z^{-2} -6 a^6-a^4 z^4+a^4 z^{-2} +a^4+2 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{11}-4 z^5 a^{11}+4 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-14 z^6 a^{10}+19 z^4 a^{10}-8 z^2 a^{10}+2 z^9 a^9-5 z^7 a^9-5 z^5 a^9+12 z^3 a^9-2 z a^9+8 z^8 a^8-35 z^6 a^8+47 z^4 a^8-27 z^2 a^8-a^8 z^{-2} +7 a^8+2 z^9 a^7-2 z^7 a^7-12 z^5 a^7+17 z^3 a^7-9 z a^7+2 a^7 z^{-1} +5 z^8 a^6-19 z^6 a^6+28 z^4 a^6-25 z^2 a^6-2 a^6 z^{-2} +11 a^6+4 z^7 a^5-10 z^5 a^5+10 z^3 a^5-7 z a^5+2 a^5 z^{-1} +2 z^6 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 2 -1 -5               3     3 -7             4 3     -1 -9           4 3       1 -11         3 4         1 -13       4 4           0 -15     2 5             3 -17   1 2               -1 -19   2                 2 -21 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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