L11a29

L11a28

L11a30

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Link Presentations

[edit Notes on L11a29's Link Presentations]

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

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) \left(5 t(2)^2-8 t(2)+5\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 22 q^{9/2}-21 q^{7/2}+14 q^{5/2}-8 q^{3/2}+q^{21/2}-4 q^{19/2}+10 q^{17/2}-15 q^{15/2}+21 q^{13/2}-24 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} + a^{-9} z^{-1} -z^5 a^{-7} +z^3 a^{-7} +z a^{-7} - a^{-7} z^{-1} -3 z^5 a^{-5} -6 z^3 a^{-5} -5 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} +z^3 a^{-3} +3 z a^{-3} +2 a^{-3} z^{-1} +z^3 a^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-6} -2 z^{10} a^{-8} -6 z^9 a^{-5} -13 z^9 a^{-7} -7 z^9 a^{-9} -8 z^8 a^{-4} -16 z^8 a^{-6} -16 z^8 a^{-8} -8 z^8 a^{-10} -6 z^7 a^{-3} -2 z^7 a^{-5} +17 z^7 a^{-7} +9 z^7 a^{-9} -4 z^7 a^{-11} -3 z^6 a^{-2} +11 z^6 a^{-4} +42 z^6 a^{-6} +48 z^6 a^{-8} +19 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +9 z^5 a^{-3} +23 z^5 a^{-5} +8 z^5 a^{-7} +3 z^5 a^{-9} +8 z^5 a^{-11} +4 z^4 a^{-2} -7 z^4 a^{-4} -37 z^4 a^{-6} -44 z^4 a^{-8} -16 z^4 a^{-10} +2 z^4 a^{-12} +2 z^3 a^{-1} -9 z^3 a^{-3} -31 z^3 a^{-5} -19 z^3 a^{-7} -3 z^3 a^{-9} -4 z^3 a^{-11} -z^2 a^{-2} -2 z^2 a^{-4} +14 z^2 a^{-6} +25 z^2 a^{-8} +9 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} +7 z a^{-3} +15 z a^{-5} +9 z a^{-7} +2 z a^{-9} +2 a^{-4} -4 a^{-6} -9 a^{-8} -4 a^{-10} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-7} z^{-1} + a^{-9} z^{-1} }[/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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   7 1   -6 16                 8 3     5 14               13 7       -6 12             11 8         3 10           11 13           2 8         10 11             -1 6       4 11               7 4     4 10                 -6 2   1 6                   5 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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