L11a92

L11a91

L11a93

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Link Presentations

[edit Notes on L11a92's Link Presentations]

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

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(2)^5+3 t(1) t(2)^4-6 t(2)^4-10 t(1) t(2)^3+11 t(2)^3+11 t(1) t(2)^2-10 t(2)^2-6 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{20}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{13}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} -6 z^3 a^7-11 z a^7-6 a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+10 z a^5+5 a^5 z^{-1} +z^5 a^3-z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+8 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-5 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^9+5 a^9 z^7-27 a^9 z^5+31 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^{10}+11 a^8 z^8-22 a^8 z^6+4 a^8 z^4+7 a^8 z^2-3 a^8+7 a^7 z^9+4 a^7 z^7-41 a^7 z^5+52 a^7 z^3-29 a^7 z+6 a^7 z^{-1} +a^6 z^{10}+14 a^6 z^8-28 a^6 z^6+15 a^6 z^4-a^6+4 a^5 z^9+8 a^5 z^7-33 a^5 z^5+41 a^5 z^3-23 a^5 z+5 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+5 a^4 z^4-a^4+6 a^3 z^7-10 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z }[/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 2                       1 1 0                     2   -2 -2                   6 1   5 -4                 8 3     -5 -6               10 5       5 -8             10 8         -2 -10           10 10           0 -12         8 11             3 -14       5 9               -4 -16     2 8                 6 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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