L11a396

L11a395

L11a397

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Link Presentations

[edit Notes on L11a396's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,14,9,13 X20,12,21,11 X12,22,13,21 X18,16,19,15 X8,18,5,17 X16,8,17,7 X14,20,15,19 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {11, -2, 4, -5, 3, -9, 6, -8, 7, -6, 9, -4, 5, -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 w^3-3 u v w^2+3 u v w-2 u v-2 u w^3+4 u w^2-4 u w+3 u-3 v w^3+4 v w^2-4 v w+2 v+2 w^3-3 w^2+3 w-2}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-8 q^6+11 q^5-14 q^4+15 q^3-14 q^2+12 q-6+5 q^{-1} - q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -z^2 a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} +3 a^2+8 a^{-2} -2 z^4-7 z^2-5 z^{-2} -9 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +5 z^9 a^{-3} +4 z^9 a^{-5} -z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +z^8+a z^7+z^7 a^{-1} -12 z^7 a^{-3} -5 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6+8 z^6 a^{-2} -11 z^6 a^{-4} -14 z^6 a^{-6} +4 z^6 a^{-8} +2 z^6-a z^5+20 z^5 a^{-3} +5 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -5 a^2 z^4-17 z^4 a^{-2} +8 z^4 a^{-4} +10 z^4 a^{-6} -6 z^4 a^{-8} -14 z^4-6 a z^3-15 z^3 a^{-1} -19 z^3 a^{-3} -5 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +9 a^2 z^2+15 z^2 a^{-2} +z^2 a^{-4} -3 z^2 a^{-6} +20 z^2+11 a z+21 z a^{-1} +13 z a^{-3} +3 z a^{-5} -7 a^2-10 a^{-2} -2 a^{-4} -14-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 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                     3   3 13                   5 1   -4 11                 6 3     3 9               8 5       -3 7             7 6         1 5           7 8           1 3         5 7             -2 1       4 10               6 -1     1 2                 -1 -3     4                   4 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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