L11a417

L11a416

L11a418

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Link Presentations

[edit Notes on L11a417's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X22,14,11,13 X20,16,21,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X14,22,15,21 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, 7, -8}, {11, -2, 3, -9, 4, -6, 5, -7, 8, -4, 9, -3}

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 v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v-2 u w+3 u-3 v^2 w^2+2 v^2 w+2 v w^2-4 v w+2 v+2 w-2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-5 q^6+8 q^5-10 q^4+11 q^3+ q^{-3} -10 q^2- q^{-2} +9 q+4 q^{-1} -5 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-6} -7 z^2+3 a^2+3 a^{-2} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +4 z^9 a^{-3} +3 z^9 a^{-5} -3 z^8 a^{-2} +4 z^8 a^{-6} +z^8+a z^7-z^7 a^{-1} -15 z^7 a^{-3} -9 z^7 a^{-5} +4 z^7 a^{-7} +a^2 z^6+8 z^6 a^{-2} -4 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} -2 a z^5+26 z^5 a^{-3} +14 z^5 a^{-5} -9 z^5 a^{-7} +z^5 a^{-9} -5 a^2 z^4-10 z^4 a^{-2} +10 z^4 a^{-4} +9 z^4 a^{-6} -7 z^4 a^{-8} -9 z^4-3 a z^3-5 z^3 a^{-1} -14 z^3 a^{-3} -6 z^3 a^{-5} +4 z^3 a^{-7} -2 z^3 a^{-9} +8 a^2 z^2+5 z^2 a^{-2} -4 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +13 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   3 1   -2 11                 5 2     3 9               5 3       -2 7             6 5         1 5           6 7           1 3         3 4             -1 1       3 7               4 -1     1 2                 -1 -3     3                   3 -5 1 1                     0 -7 1                       1

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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