L11a296

L11a295

L11a297

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Link Presentations

[edit Notes on L11a296's Link Presentations]

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

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)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3+3 t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{9}{q^{9/2}}-q^{7/2}+\frac{11}{q^{7/2}}+3 q^{5/2}-\frac{14}{q^{5/2}}-6 q^{3/2}+\frac{14}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{5}{q^{11/2}}+9 \sqrt{q}-\frac{13}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+5 a^3 z^5+4 a z^5-z^5 a^{-1} -4 a^5 z^3+10 a^3 z^3+4 a z^3-3 z^3 a^{-1} -5 a^5 z+9 a^3 z-2 a z-2 z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-3 a^6 z^8-3 a^4 z^8-5 a^2 z^8-5 z^8-2 a^7 z^7+10 a^5 z^7+19 a^3 z^7+2 a z^7-5 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+14 a^4 z^6+18 a^2 z^6-3 z^6 a^{-2} +11 z^6+6 a^7 z^5-21 a^5 z^5-35 a^3 z^5+5 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +4 a^8 z^4-10 a^6 z^4-23 a^4 z^4-25 a^2 z^4+6 z^4 a^{-2} -10 z^4-4 a^7 z^3+27 a^5 z^3+35 a^3 z^3-8 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} -4 a^8 z^2+7 a^6 z^2+18 a^4 z^2+11 a^2 z^2-z^2 a^{-2} +3 z^2-10 a^5 z-17 a^3 z-3 a z+4 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   4 1   3 2                 5 2     -3 0               8 4       4 -2             8 7         -1 -4           6 6           0 -6         5 8             3 -8       4 6               -2 -10     1 5                 4 -12   1 4                   -3 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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