L11a508

L11a507

L11a509

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Link Presentations

[edit Notes on L11a508's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X4,21,5,22 X14,6,15,5 X22,15,13,16 X6,14,1,13
Gauss code	{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -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{(w-1) \left(u^2 v^2 w+u^2 \left(-v^2\right)+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 u v^2-2 u v w^2+3 u v w-2 u v+2 u w^2-2 u w+v^2 w-v^2+v w^2-2 v w+v-w^2+w\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+4 q^6-7 q^5+13 q^4+ q^{-4} -16 q^3-4 q^{-3} +20 q^2+8 q^{-2} -19 q-13 q^{-1} +18 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-z^4 a^{-2} +2 z^4 a^{-4} -2 z^4+a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -2 z^2+2 a^{-2} -2 a^{-4} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-7} -3 z^5 a^{-7} +2 z^3 a^{-7} +4 z^8 a^{-6} -15 z^6 a^{-6} +17 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +5 z^9 a^{-5} -15 z^7 a^{-5} +10 z^5 a^{-5} +2 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +7 z^8 a^{-4} -41 z^6 a^{-4} +a^4 z^4+47 z^4 a^{-4} -14 z^2 a^{-4} +2 a^{-4} z^{-2} -3 a^{-4} +12 z^9 a^{-3} -30 z^7 a^{-3} +4 a^3 z^5+14 z^5 a^{-3} -2 a^3 z^3+2 z^3 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +14 z^8 a^{-2} +8 a^2 z^6-49 z^6 a^{-2} -7 a^2 z^4+42 z^4 a^{-2} +2 a^2 z^2-11 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +7 z^9 a^{-1} +11 a z^7-3 z^7 a^{-1} -13 a z^5-16 z^5 a^{-1} +4 a z^3+10 z^3 a^{-1} +11 z^8-15 z^6+4 z^4 }[/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 15                       1 -1 13                     3   3 11                   4 1   -3 9                 9 3     6 7               8 5       -3 5             12 8         4 3           9 10           1 1         9 10             -1 -1       5 10               5 -3     3 8                 -5 -5   1 5                   4 -7   3                     -3 -9 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=1 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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