L11a294

L11a293

L11a295

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Link Presentations

[edit Notes on L11a294's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X22,20,9,19 X18,7,19,8 X16,5,17,6 X4,15,5,16 X6,17,7,18 X14,22,15,21 X20,14,21,13 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -6, 5, -7, 4, -11}, {10, -1, 11, -2, 9, -8, 6, -5, 7, -4, 3, -9, 8, -3}

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)^2 t(2)^5+t(1)^3 t(2)^4-3 t(1)^2 t(2)^4+2 t(1) t(2)^4-2 t(1)^3 t(2)^3+5 t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-4 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+5 z a^7+2 a^7 z^{-1} -z^7 a^5-5 z^5 a^5-10 z^3 a^5-10 z a^5-3 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-4 z^3 a^3+a^3 z^{-1} +z^5 a+3 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+6 z^5 a^9-4 z^3 a^9+2 z a^9-3 z^8 a^8+5 z^6 a^8-4 z^4 a^8+2 z^2 a^8-3 z^9 a^7+9 z^7 a^7-20 z^5 a^7+21 z^3 a^7-10 z a^7+2 a^7 z^{-1} -z^{10} a^6-3 z^8 a^6+16 z^6 a^6-30 z^4 a^6+18 z^2 a^6-3 a^6-6 z^9 a^5+22 z^7 a^5-39 z^5 a^5+34 z^3 a^5-16 z a^5+3 a^5 z^{-1} -z^{10} a^4-4 z^8 a^4+23 z^6 a^4-36 z^4 a^4+20 z^2 a^4-3 a^4-3 z^9 a^3+7 z^7 a^3-2 z^5 a^3-z^3 a^3-z a^3+a^3 z^{-1} -4 z^8 a^2+13 z^6 a^2-11 z^4 a^2+4 z^2 a^2-a^2-3 z^7 a+10 z^5 a-7 z^3 a+z a-z^6+3 z^4-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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   3 1   -2 -2                 5 2     3 -4               7 4       -3 -6             6 4         2 -8           6 7           1 -10         5 6             -1 -12       3 6               3 -14     2 5                 -3 -16     3                   3 -18 1 2                     -1 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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