L10a153

L10a152

L10a154

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Link Presentations

[edit Notes on L10a153's Link Presentations]

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

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{(w-1) \left(2 u v^2 w^2-u v w^2+2 u v w-u w+u+v^2 w^3-v^2 w^2+2 v w^2-v w+2 w\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -9 q^{-8} +9 q^{-9} -7 q^{-10} +5 q^{-11} -2 q^{-12} + q^{-13} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^2+a^{12} z^{-2} +3 a^{12}-3 a^{10} z^4-11 a^{10} z^2-2 a^{10} z^{-2} -10 a^{10}+2 a^8 z^6+9 a^8 z^4+12 a^8 z^2+a^8 z^{-2} +6 a^8+a^6 z^6+4 a^6 z^4+4 a^6 z^2+a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-2 a^{15} z^3+3 a^{14} z^6-3 a^{14} z^4+a^{14} z^2+3 a^{13} z^7-2 a^{13} z^5+3 a^{12} z^8-6 a^{12} z^6+11 a^{12} z^4-11 a^{12} z^2-a^{12} z^{-2} +6 a^{12}+a^{11} z^9+4 a^{11} z^7-17 a^{11} z^5+24 a^{11} z^3-13 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-21 a^{10} z^6+34 a^{10} z^4-32 a^{10} z^2-2 a^{10} z^{-2} +14 a^{10}+a^9 z^9+3 a^9 z^7-19 a^9 z^5+25 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+15 a^8 z^4-14 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+3 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^2-a^6 }[/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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 3     3 -11               3 2     -1 -13             6 3       3 -15           4 4         0 -17         5 5           0 -19       3 5             2 -21     2 4               -2 -23     3                 3 -25 1 2                   -1 -27 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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