L11a488

L11a487

L11a489

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Link Presentations

[edit Notes on L11a488's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X22,18,19,17 X20,14,21,13 X14,20,15,19 X18,22,5,21 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -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{(u-1) (w-1)^2 \left(2 v w^2-v w+v+w^3-w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11}+3 q^{10}-8 q^9+14 q^8-18 q^7+21 q^6-20 q^5+19 q^4-12 q^3+8 q^2-3 q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-10} - a^{-10} z^{-2} -2 a^{-10} +3 z^4 a^{-8} +8 z^2 a^{-8} +4 a^{-8} z^{-2} +9 a^{-8} -2 z^6 a^{-6} -7 z^4 a^{-6} -12 z^2 a^{-6} -5 a^{-6} z^{-2} -13 a^{-6} -z^6 a^{-4} -z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} z^{-2} +5 a^{-4} +z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +9 z^9 a^{-7} +5 z^9 a^{-9} +5 z^8 a^{-4} +16 z^8 a^{-6} +19 z^8 a^{-8} +8 z^8 a^{-10} +3 z^7 a^{-3} -5 z^7 a^{-7} +4 z^7 a^{-9} +6 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -51 z^6 a^{-6} -56 z^6 a^{-8} -14 z^6 a^{-10} +3 z^6 a^{-12} -7 z^5 a^{-3} -18 z^5 a^{-5} -29 z^5 a^{-7} -28 z^5 a^{-9} -9 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +10 z^4 a^{-4} +64 z^4 a^{-6} +71 z^4 a^{-8} +16 z^4 a^{-10} -4 z^4 a^{-12} +4 z^3 a^{-3} +24 z^3 a^{-5} +50 z^3 a^{-7} +39 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} -11 z^2 a^{-4} -48 z^2 a^{-6} -48 z^2 a^{-8} -13 z^2 a^{-10} +z^2 a^{-12} -17 z a^{-5} -35 z a^{-7} -23 z a^{-9} -4 z a^{-11} +z a^{-13} - a^{-2} +8 a^{-4} +22 a^{-6} +19 a^{-8} +5 a^{-10} +5 a^{-5} z^{-1} +9 a^{-7} z^{-1} +5 a^{-9} z^{-1} + a^{-11} z^{-1} -2 a^{-4} z^{-2} -5 a^{-6} z^{-2} -4 a^{-8} z^{-2} - a^{-10} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     2   2 19                   6 1   -5 17                 8 2     6 15               10 6       -4 13             11 8         3 11           11 12           1 9         8 9             -1 7       4 11               7 5     4 8                 -4 3   1 6                   5 1   2                     -2 -1 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=5 }[/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}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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