L11a276

L11a275

L11a277

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Link Presentations

[edit Notes on L11a276's Link Presentations]

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

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)^3 t(2)^5-2 t(1)^3 t(2)^4+2 t(1)^2 t(2)^4+2 t(1)^3 t(2)^3-3 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)+1}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{7}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{7}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^9+a^7 z^7-8 a^5 z^7+a^3 z^7+6 a^7 z^5-24 a^5 z^5+6 a^3 z^5+12 a^7 z^3-34 a^5 z^3+11 a^3 z^3+10 a^7 z-21 a^5 z+7 a^3 z+2 a^7 z^{-1} -3 a^5 z^{-1} +a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{13}+z a^{13}-2 z^4 a^{12}+z^2 a^{12}-3 z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-3 z^6 a^{10}+3 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+5 z^5 a^9-2 z^3 a^9+z a^9-3 z^8 a^8+8 z^6 a^8-6 z^4 a^8+4 z^2 a^8-3 z^9 a^7+13 z^7 a^7-22 z^5 a^7+24 z^3 a^7-12 z a^7+2 a^7 z^{-1} -z^{10} a^6+16 z^6 a^6-30 z^4 a^6+19 z^2 a^6-3 a^6-5 z^9 a^5+28 z^7 a^5-53 z^5 a^5+48 z^3 a^5-23 z a^5+3 a^5 z^{-1} -z^{10} a^4+2 z^8 a^4+11 z^6 a^4-30 z^4 a^4+20 z^2 a^4-3 a^4-2 z^9 a^3+12 z^7 a^3-23 z^5 a^3+18 z^3 a^3-7 z a^3+a^3 z^{-1} -z^8 a^2+6 z^6 a^2-11 z^4 a^2+7 z^2 a^2-a^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 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 3 1     2 -6               3 3       0 -8             4 2         2 -10           3 3           0 -12         3 4             -1 -14       2 3               1 -16     2 3                 -1 -18     2                   2 -20 1 2                     -1 -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=-6 }[/math] [math]\displaystyle{ i=-4 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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