L11n416

L11n415

L11n417

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Link Presentations

[edit Notes on L11n416's Link Presentations]

Planar diagram presentation	X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, -9, 11, 8, -7, -10, 9, -11, 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^2 v w^3-u^2 v w^2+u^2 v w-u^2 v-u^2 w^3+u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-2 u v w^3+2 u v w^2-2 u v w+2 u v+u w^3-u w^2+u w-u+v^2+v w^3-v w^2+v w-v}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{-9} +4 q^{-8} -6 q^{-7} +9 q^{-6} -8 q^{-5} +9 q^{-4} -6 q^{-3} +5 q^{-2} -2 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+a^8 z^4+4 a^8 z^2+4 a^8 z^{-2} +6 a^8-a^6 z^6-4 a^6 z^4-7 a^6 z^2-5 a^6 z^{-2} -9 a^6-a^4 z^6-3 a^4 z^4-a^4 z^2+2 a^4 z^{-2} +2 a^4+a^2 z^4+3 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +a^{10} z^6+2 a^{10} z^4-5 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+4 a^9 z^7-15 a^9 z^5+30 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +4 a^8 z^8-16 a^8 z^6+34 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9+4 a^7 z^7-23 a^7 z^5+43 a^7 z^3-33 a^7 z+9 a^7 z^{-1} +6 a^6 z^8-22 a^6 z^6+36 a^6 z^4-36 a^6 z^2-5 a^6 z^{-2} +20 a^6+a^5 z^9+2 a^5 z^7-14 a^5 z^5+19 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-4 a^4 z^6-4 a^4 z^2-2 a^4 z^{-2} +6 a^4+2 a^3 z^7-6 a^3 z^5+3 a^3 z^3+a^3 z+a^2 z^6-4 a^2 z^4+5 a^2 z^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               4 1   3 -5             4 3     -1 -7           5 2       3 -9         4 5         1 -11       5 4           1 -13     1 4             3 -15   3 5               -2 -17   2                 2 -19 2                   -2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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