L11a398

L11a397

L11a399

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Link Presentations

[edit Notes on L11a398's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {8, -2, 4, -5, 10, -9, 7, -6, 11, -8, 9, -7}

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{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(2) t(3)^3-2 t(2) t(3)^2+2 t(3)^2+2 t(2) t(3)-2 t(3)+1\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-5 q^{-5} +4 q^4+10 q^{-4} -9 q^3-17 q^{-3} +17 q^2+23 q^{-2} -21 q-25 q^{-1} +27 }[/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^4-6 a^2 z^4-3 z^4 a^{-2} +10 z^4+a^4 z^2-4 a^2 z^2-3 z^2 a^{-2} +6 z^2-a^4+4 a^2+ a^{-2} -4-a^4 z^{-2} +4 a^2 z^{-2} +2 a^{-2} z^{-2} -5 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-a^6 z^4+5 a^5 z^7-10 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-z^3 a^{-5} +a^5 z-a^5 z^{-1} +9 a^4 z^8-19 a^4 z^6+4 z^6 a^{-4} +12 a^4 z^4-5 z^4 a^{-4} -4 a^4 z^2+2 z^2 a^{-4} +a^4 z^{-2} +7 a^3 z^9-a^3 z^7+8 z^7 a^{-3} -25 a^3 z^5-10 z^5 a^{-3} +17 a^3 z^3+4 z^3 a^{-3} +5 a^3 z-5 a^3 z^{-1} +2 a^2 z^{10}+22 a^2 z^8+11 z^8 a^{-2} -63 a^2 z^6-17 z^6 a^{-2} +53 a^2 z^4+14 z^4 a^{-2} -18 a^2 z^2-5 z^2 a^{-2} +4 a^2 z^{-2} +2 a^{-2} z^{-2} -2 a^2-3 a^{-2} +15 a z^9+8 z^9 a^{-1} -15 a z^7-z^7 a^{-1} -19 a z^5-15 z^5 a^{-1} +17 a z^3+10 z^3 a^{-1} +9 a z+5 z a^{-1} -9 a z^{-1} -5 a^{-1} z^{-1} +2 z^{10}+24 z^8-64 z^6+59 z^4-21 z^2+5 z^{-2} -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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 11 3     8 3               12 8       -4 1             15 9         6 -1           12 14           2 -3         11 13             -2 -5       6 12               6 -7     4 11                 -7 -9   1 6                   5 -11   4                     -4 -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{ 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}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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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