L11n415

L11n414

L11n416

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Link Presentations

[edit Notes on L11n415's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21
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{2 u^2 v w^2-2 u^2 v w+u^2 v-u^2 w^2+u^2 w+2 u v^2 w^2-2 u v^2 w+u v^2+u v w^3-4 u v w^2+4 u v w-u v-u w^3+2 u w^2-2 u w-v^2 w^2+v^2 w-v w^3+2 v w^2-2 v w}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+9 q^5-11 q^4+12 q^3-10 q^2+9 q+3 q^{-1} -4 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-8} +3 z^2 a^{-6} - a^{-6} z^{-2} +2 a^{-6} -2 z^4 a^{-4} -z^2 a^{-4} +4 a^{-4} z^{-2} +4 a^{-4} -3 z^4 a^{-2} -8 z^2 a^{-2} -5 a^{-2} z^{-2} -10 a^{-2} +3 z^2+2 z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +4 z^8 a^{-2} +7 z^8 a^{-4} +3 z^8 a^{-6} +3 z^7 a^{-1} +7 z^7 a^{-3} +8 z^7 a^{-5} +4 z^7 a^{-7} -14 z^6 a^{-2} -17 z^6 a^{-4} +3 z^6 a^{-8} -9 z^5 a^{-1} -28 z^5 a^{-3} -26 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +32 z^4 a^{-2} +23 z^4 a^{-4} -9 z^4 a^{-6} -6 z^4 a^{-8} +6 z^4+18 z^3 a^{-1} +48 z^3 a^{-3} +33 z^3 a^{-5} +z^3 a^{-7} -2 z^3 a^{-9} -38 z^2 a^{-2} -19 z^2 a^{-4} +6 z^2 a^{-6} +3 z^2 a^{-8} -16 z^2-19 z a^{-1} -35 z a^{-3} -19 z a^{-5} -2 z a^{-7} +z a^{-9} +22 a^{-2} +13 a^{-4} - a^{-8} +11+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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             5 2     3 9           7 5       -2 7         5 4         1 5       5 7           2 3     4 5             -1 1   1 6               5 -1 2 3                 -1 -3 3                   3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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