L10a106

L10a105

L10a107

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Link Presentations

[edit Notes on L10a106's Link Presentations]

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

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(2)^3 t(1)^3-2 t(2)^2 t(1)^3+t(2) t(1)^3-3 t(2)^3 t(1)^2+7 t(2)^2 t(1)^2-5 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-5 t(2)^2 t(1)+7 t(2) t(1)-3 t(1)+t(2)^2-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{13}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-2 a^7 z+2 a^5 z^5+6 a^5 z^3+5 a^5 z+a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-5 a^3 z-a^3 z^{-1} +a z^5+2 a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-z a^9-5 z^6 a^8+4 z^4 a^8-z^2 a^8-7 z^7 a^7+10 z^5 a^7-8 z^3 a^7+z a^7-6 z^8 a^6+7 z^6 a^6-2 z^4 a^6-z^2 a^6-2 z^9 a^5-10 z^7 a^5+33 z^5 a^5-30 z^3 a^5+10 z a^5-a^5 z^{-1} -11 z^8 a^4+24 z^6 a^4-12 z^4 a^4+z^2 a^4+a^4-2 z^9 a^3-7 z^7 a^3+31 z^5 a^3-27 z^3 a^3+9 z a^3-a^3 z^{-1} -5 z^8 a^2+11 z^6 a^2-3 z^4 a^2-z^2 a^2-4 z^7 a+11 z^5 a-8 z^3 a+z a-z^6+2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   3   3 0                 4 1   -3 -2               7 3     4 -4             7 5       -2 -6           7 6         1 -8         6 7           1 -10       5 7             -2 -12     2 7               5 -14   1 4                 -3 -16   2                   2 -18 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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