L11a277

L11a276

L11a278

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Link Presentations

[edit Notes on L11a277's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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-t(1)^2 t(2)^5-t(1)^3 t(2)^4+t(1)^2 t(2)^4-t(1) t(2)^4+t(1)^3 t(2)^3-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^7\right)-6 a^9 z^5-11 a^9 z^3-7 a^9 z-a^9 z^{-1} +a^7 z^9+8 a^7 z^7+23 a^7 z^5+30 a^7 z^3+18 a^7 z+3 a^7 z^{-1} -a^5 z^7-7 a^5 z^5-16 a^5 z^3-13 a^5 z-2 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{16}-2 z^3 a^{15}-3 z^4 a^{14}+2 z^2 a^{14}-4 z^5 a^{13}+7 z^3 a^{13}-2 z a^{13}-4 z^6 a^{12}+9 z^4 a^{12}-2 z^2 a^{12}-4 z^7 a^{11}+12 z^5 a^{11}-5 z^3 a^{11}-z a^{11}-4 z^8 a^{10}+17 z^6 a^{10}-19 z^4 a^{10}+7 z^2 a^{10}-a^{10}-3 z^9 a^9+15 z^7 a^9-22 z^5 a^9+13 z^3 a^9-6 z a^9+a^9 z^{-1} -z^{10} a^8+2 z^8 a^8+12 z^6 a^8-34 z^4 a^8+23 z^2 a^8-3 a^8-4 z^9 a^7+27 z^7 a^7-61 z^5 a^7+56 z^3 a^7-22 z a^7+3 a^7 z^{-1} -z^{10} a^6+6 z^8 a^6-9 z^6 a^6-3 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5+8 z^7 a^5-23 z^5 a^5+29 z^3 a^5-15 z a^5+2 a^5 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -2                       1 1 -4                         0 -6                   3 1   2 -8                 1 1     0 -10               3 2       1 -12             2 1         -1 -14           3 3           0 -16         2 3             1 -18       2 2               0 -20     1 2                 1 -22   1 2                   -1 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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