L11a355

L11a354

L11a356

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Link Presentations

[edit Notes on L11a355's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,7,19,8 X14,3,15,4 X16,5,17,6 X20,10,21,9 X22,19,11,20 X4,15,5,16 X6,17,7,18 X8,22,9,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -7, 4, -8, 2, -9, 5, -11}, {10, -1, 11, -3, 7, -4, 8, -2, 6, -5, 9, -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)+t(2)-1) (t(2) t(1)-t(1)-t(2)) \left(t(1)^2 t(2)^2+t(1) t(2)+1\right)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{7}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-4 z^3 a^9-3 z a^9+z^7 a^7+5 z^5 a^7+8 z^3 a^7+6 z a^7+a^7 z^{-1} +z^7 a^5+4 z^5 a^5+2 z^3 a^5-3 z a^5-a^5 z^{-1} -z^5 a^3-4 z^3 a^3-3 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+2 z^2 a^{14}-2 z^5 a^{13}+4 z^3 a^{13}-2 z a^{13}-2 z^6 a^{12}+2 z^4 a^{12}-2 z^7 a^{11}+2 z^5 a^{11}-3 z a^{11}-2 z^8 a^{10}+4 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}-2 z^9 a^9+7 z^7 a^9-13 z^5 a^9+10 z^3 a^9-2 z a^9-z^{10} a^8+2 z^8 a^8-3 z^4 a^8+4 z^2 a^8-4 z^9 a^7+18 z^7 a^7-30 z^5 a^7+26 z^3 a^7-9 z a^7+a^7 z^{-1} -z^{10} a^6+2 z^8 a^6+3 z^6 a^6-5 z^4 a^6+3 z^2 a^6-a^6-2 z^9 a^5+8 z^7 a^5-8 z^5 a^5+5 z^3 a^5-5 z a^5+a^5 z^{-1} -2 z^8 a^4+9 z^6 a^4-10 z^4 a^4+2 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3 }[/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 0                       1 1 -2                     1   -1 -4                   3 1   2 -6                 3 2     -1 -8               4 2       2 -10             4 3         -1 -12           4 4           0 -14         3 4             1 -16       3 4               -1 -18     1 4                 3 -20   1 2                   -1 -22   1                     1 -24 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=-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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\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 }[/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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