L11a376

L11a375

L11a377

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Link Presentations

[edit Notes on L11a376's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,3,15,4 X16,5,17,6 X22,7,11,8 X18,10,19,9 X20,18,21,17 X10,20,1,19 X8,11,9,12 X6,21,7,22 X4,13,5,14 X2,15,3,16
Gauss code	{1, -11, 2, -10, 3, -9, 4, -8, 5, -7}, {8, -1, 10, -2, 11, -3, 6, -5, 7, -6, 9, -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) t(1)-t(1)+1) (t(1) t(2)-t(2)+1) \left(t(2) t(1)^2+t(2)^2 t(1)-t(2) t(1)+t(1)+t(2)\right)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{14}{q^{9/2}}-\frac{15}{q^{7/2}}+\frac{12}{q^{5/2}}+q^{3/2}-\frac{10}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{12}{q^{11/2}}-3 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+5 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-10 z^3 a^5-8 z a^5-a^5 z^{-1} -z^7 a^3-4 z^5 a^3-5 z^3 a^3-2 z a^3+z^5 a+3 z^3 a+2 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+5 z^5 a^9-2 z^3 a^9+z a^9-4 z^8 a^8+10 z^6 a^8-15 z^4 a^8+9 z^2 a^8-3 z^9 a^7+5 z^7 a^7-6 z^5 a^7+3 z^3 a^7-4 z a^7+a^7 z^{-1} -z^{10} a^6-5 z^8 a^6+19 z^6 a^6-29 z^4 a^6+14 z^2 a^6-a^6-6 z^9 a^5+15 z^7 a^5-18 z^5 a^5+13 z^3 a^5-8 z a^5+a^5 z^{-1} -z^{10} a^4-5 z^8 a^4+18 z^6 a^4-18 z^4 a^4+7 z^2 a^4-3 z^9 a^3+4 z^7 a^3+3 z^5 a^3-2 z^3 a^3+z a^3-4 z^8 a^2+10 z^6 a^2-5 z^4 a^2+z^2 a^2-3 z^7 a+9 z^5 a-7 z^3 a+2 z a-z^6+3 z^4-2 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                   4 1   -3 -2                 6 2     4 -4               7 5       -2 -6             8 5         3 -8           6 7           1 -10         6 8             -2 -12       3 6               3 -14     2 6                 -4 -16   1 4                   3 -18   1                     -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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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