L10a124

L10a123

L10a125

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Link Presentations

[edit Notes on L10a124's Link Presentations]

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

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 v w^3-3 u v w^2+3 u v w-u v-u w^3+4 u w^2-4 u w+2 u-2 v w^3+4 v w^2-4 v w+v+w^3-3 w^2+3 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+4 q-7+11 q^{-1} -11 q^{-2} +14 q^{-3} -11 q^{-4} +9 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^8 z^{-2} -a^8+3 a^6 z^2+4 a^6 z^{-2} +6 a^6-3 a^4 z^4-8 a^4 z^2-5 a^4 z^{-2} -10 a^4+a^2 z^6+3 a^2 z^4+5 a^2 z^2+2 a^2 z^{-2} +5 a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+3 z a^9-a^9 z^{-1} +2 z^6 a^8-4 z^4 a^8+3 z^2 a^8+a^8 z^{-2} -2 a^8+2 z^7 a^7+2 z^5 a^7-12 z^3 a^7+13 z a^7-5 a^7 z^{-1} +2 z^8 a^6+3 z^6 a^6-11 z^4 a^6+14 z^2 a^6+4 a^6 z^{-2} -10 a^6+z^9 a^5+5 z^7 a^5-4 z^5 a^5-12 z^3 a^5+21 z a^5-9 a^5 z^{-1} +6 z^8 a^4-3 z^6 a^4-12 z^4 a^4+19 z^2 a^4+5 a^4 z^{-2} -14 a^4+z^9 a^3+9 z^7 a^3-16 z^5 a^3+11 z a^3-5 a^3 z^{-1} +4 z^8 a^2-12 z^4 a^2+11 z^2 a^2+2 a^2 z^{-2} -7 a^2+6 z^7 a-10 z^5 a+2 z^3 a+4 z^6-7 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} }[/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 3 χ 5                     1 -1 3                   3   3 1                 4 1   -3 -1               7 3     4 -3             7 7       0 -5           7 4         3 -7         4 7           3 -9       5 7             -2 -11     1 5               4 -13   1 4                 -3 -15   1                   1 -17 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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