L11a298

L11a297

L11a299

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Link Presentations

[edit Notes on L11a298's Link Presentations]

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

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)-1) (t(2)-1) \left(2 t(2) t(1)^2+2 t(2)^2 t(1)-t(2) t(1)+2 t(1)+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+3 q^{7/2}-6 q^{5/2}+8 q^{3/2}-10 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^5-2 z a^5+z^5 a^3+2 z^3 a^3+2 z^5 a+6 z^3 a+5 z a+a z^{-1} +z^5 a^{-1} +z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-2 a^5 z^9-5 a^3 z^9-3 a z^9-a^6 z^8+a^4 z^8-4 a^2 z^8-6 z^8+11 a^5 z^7+19 a^3 z^7-8 z^7 a^{-1} +6 a^6 z^6+11 a^4 z^6+23 a^2 z^6-8 z^6 a^{-2} +10 z^6-20 a^5 z^5-17 a^3 z^5+23 a z^5+14 z^5 a^{-1} -6 z^5 a^{-3} -12 a^6 z^4-22 a^4 z^4-22 a^2 z^4+11 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4+14 a^5 z^3-26 a z^3-6 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +8 a^6 z^2+9 a^4 z^2+4 a^2 z^2-4 z^2 a^{-2} -z^2-4 a^5 z+10 a z+4 z a^{-1} -2 z a^{-3} +1-a z^{-1} - a^{-1} 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 10                       1 1 8                     2   -2 6                   4 1   3 4                 4 2     -2 2               6 4       2 0             7 6         -1 -2           4 4           0 -4         4 7             3 -6       3 4               -1 -8     1 4                 3 -10   1 3                   -2 -12   1                     1 -14 1                       -1

Integral Khovanov Homology (db, data source)

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