L10a157

L10a156

L10a158

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Link Presentations

[edit Notes on L10a157's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,13,10 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16
Gauss code	{1, -4, 3, -8, 9, -10}, {5, -1, 6, -3, 2, -7}, {8, -9, 10, -5, 7, -2, 4, -6}

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(u^2 v^2 w+u^2 v w^2-u^2 v w-u^2 w^2-u v^2 w+u v^2-u v w^2+u v w-u v+u w^2-u w-v^2-v w+v+w\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-6} +3 q^{-5} -q^4-5 q^{-4} +3 q^3+8 q^{-3} -4 q^2-9 q^{-2} +8 q+10 q^{-1} -8 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-z^4 a^{-2} +3 z^4-2 a^4 z^2+2 a^2 z^2-2 z^2 a^{-2} +z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^3+3 a^6 z^4-a^6 z^2+5 a^5 z^5-3 a^5 z^3+7 a^4 z^6-10 a^4 z^4+4 a^4 z^2+7 a^3 z^7+z^7 a^{-3} -13 a^3 z^5-4 z^5 a^{-3} +5 a^3 z^3+4 z^3 a^{-3} +5 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-14 z^6 a^{-2} -a^2 z^4+20 z^4 a^{-2} +3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +2 a z^9+2 z^9 a^{-1} +a z^7-5 z^7 a^{-1} -16 a z^5-2 z^5 a^{-1} +11 a z^3+6 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +8 z^8-30 z^6+32 z^4-10 z^2+2 z^{-2} -3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                     1 -1 7                   2   2 5                 2 1   -1 3               6 2     4 1             3 3       0 -1           7 5         2 -3         4 5           1 -5       4 5             -1 -7     2 5               3 -9   1 3                 -2 -11   2                   2 -13 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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