L11a469

L11a468

L11a470

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Link Presentations

[edit Notes on L11a469's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X16,8,17,7 X8,16,9,15 X22,17,15,18 X12,21,13,22 X20,13,21,14 X14,19,5,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 6, -3, 9, -8, 7, -6}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9}

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-u v^2+2 u v w^2-4 u v w+u v-2 u w^2+u w+v^2 w^2-2 v^2 w+v w^3-4 v w^2+2 v w-w^3+2 w^2\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +3 q^{-9} -6 q^{-8} +11 q^{-7} -13 q^{-6} +17 q^{-5} -16 q^{-4} +15 q^{-3} -11 q^{-2} +q+7 q^{-1} -3 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+3 a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^4-a^6 z^2-2 a^6 z^{-2} -2 a^6-3 a^4 z^4-3 a^4 z^2+a^4 z^{-2} -a^4-a^2 z^4+2 a^2 z^2+2 a^2+z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-4 a^{11} z^5+5 a^{11} z^3-2 a^{11} z+3 a^{10} z^8-12 a^{10} z^6+16 a^{10} z^4-9 a^{10} z^2+3 a^{10}+3 a^9 z^9-6 a^9 z^7-7 a^9 z^5+18 a^9 z^3-7 a^9 z+a^8 z^{10}+8 a^8 z^8-38 a^8 z^6+51 a^8 z^4-34 a^8 z^2-a^8 z^{-2} +12 a^8+7 a^7 z^9-11 a^7 z^7-13 a^7 z^5+25 a^7 z^3-14 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-38 a^6 z^6+39 a^6 z^4-28 a^6 z^2-2 a^6 z^{-2} +12 a^6+4 a^5 z^9+4 a^5 z^7-24 a^5 z^5+22 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-6 a^4 z^6-4 a^4 z^4+4 a^4 z^2-a^4 z^{-2} +2 a^4+8 a^3 z^7-11 a^3 z^5+8 a^3 z^3-a^3 z+6 a^2 z^6-7 a^2 z^4+6 a^2 z^2-2 a^2+3 a z^5-2 a z^3+z^4-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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 7 3     -4 -5               8 4       4 -7             9 8         -1 -9           8 7           1 -11         6 10             4 -13       5 7               -2 -15     2 7                 5 -17   1 4                   -3 -19   2                     2 -21 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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