L11n418

L11n417

L11n419

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Link Presentations

[edit Notes on L11n418's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X3,10,4,11 X2,18,3,17 X18,9,19,10 X11,20,12,21 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X21,12,22,7
Gauss code	{1, -4, -3, 10, -7, 9}, {-2, -1, 5, 3, -6, 11}, {-9, 7, -8, 2, 4, -5, -10, 6, -11, 8}

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 w^3-2 u^2 v w^2+u^2 v w+u^2 w^2-u^2 w+u v^2 w-u v w^3+2 u v w^2-2 u v w+u v-u w^2+v^2 w^2-v^2 w-v w^2+2 v w-v}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -1+3 q^{-1} -4 q^{-2} +7 q^{-3} -6 q^{-4} +7 q^{-5} -5 q^{-6} +4 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^4-7 a^6 z^2-2 a^6 z^{-2} -8 a^6+a^4 z^6+5 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +7 a^4-a^2 z^4-2 a^2 z^2-a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-4 a^{10} z^4+4 a^{10} z^2-a^{10}+2 a^9 z^7-7 a^9 z^5+5 a^9 z^3+2 a^8 z^8-6 a^8 z^6+4 a^8 z^4-4 a^8 z^2-a^8 z^{-2} +4 a^8+a^7 z^9-a^7 z^7-4 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-16 a^6 z^6+28 a^6 z^4-27 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^9-2 a^5 z^7+a^5 z^5+6 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+23 a^4 z^4-23 a^4 z^2-a^4 z^{-2} +10 a^4+a^3 z^7-2 a^3 z^5+6 a^3 z^3-3 a^3 z+3 a^2 z^4-4 a^2 z^2+2 a^2+a z^3-a z }[/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                 2   2 -3               3 2   -1 -5             4 1     3 -7           3 4       1 -9         4 3         1 -11       2 4           2 -13     2 3             -1 -15   1 3               2 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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