L11a428

L11a427

L11a429

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Link Presentations

[edit Notes on L11a428's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,7,21,8 X8,19,9,20 X10,13,5,14 X16,22,17,21 X18,12,19,11 X22,18,11,17 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -6}, {8, -2, 6, -3, 11, -7, 9, -8, 5, -4, 7, -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{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-2 u v^2 w+u v^2-2 u v w^4+5 u v w^3-5 u v w^2+4 u v w-2 u v-2 u w^3+2 u w^2-2 u w+u-v^2 w^4+2 v^2 w^3-2 v^2 w^2+2 v^2 w+2 v w^4-4 v w^3+5 v w^2-5 v w+2 v-w^4+2 w^3-2 w^2+2 w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-3 q^{-5} +4 q^4+8 q^{-4} -9 q^3-12 q^{-3} +14 q^2+19 q^{-2} -19 q-20 q^{-1} +22 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +10 z^4+3 a^4 z^2-12 a^2 z^2-3 z^2 a^{-2} +11 z^2+3 a^4-10 a^2-2 a^{-2} +9+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+2 a^6 z^2+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +3 a^5 z^3-z^3 a^{-5} +6 a^4 z^8-18 a^4 z^6+4 z^6 a^{-4} +24 a^4 z^4-5 z^4 a^{-4} -22 a^4 z^2-2 a^4 z^{-2} +10 a^4+5 a^3 z^9-6 a^3 z^7+8 z^7 a^{-3} -9 a^3 z^5-13 z^5 a^{-3} +20 a^3 z^3+6 z^3 a^{-3} -16 a^3 z-2 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +2 a^2 z^{10}+10 a^2 z^8+10 z^8 a^{-2} -41 a^2 z^6-19 z^6 a^{-2} +59 a^2 z^4+16 z^4 a^{-2} -46 a^2 z^2-8 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +20 a^2+2 a^{-2} +12 a z^9+7 z^9 a^{-1} -23 a z^7-6 z^7 a^{-1} +5 a z^5-7 z^5 a^{-1} +26 a z^3+16 z^3 a^{-1} -27 a z-13 z a^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} +2 z^{10}+14 z^8-45 z^6+53 z^4-30 z^2-4 z^{-2} +13 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 8 3     5 3               11 6       -5 1             11 8         3 -1           11 13           2 -3         8 9             -1 -5       5 12               7 -7     3 7                 -4 -9     5                   5 -11 1 3                     -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=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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