L11a295

L11a294

L11a296

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Link Presentations

[edit Notes on L11a295's Link Presentations]

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

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{(u+v) \left(u^2 v^3-2 u^2 v^2+2 u^2 v+u v^4-4 u v^3+6 u v^2-4 u v+u+2 v^3-2 v^2+v\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^7+2 z a^7+2 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-4 z a^5-3 a^5 z^{-1} -2 z^5 a^3-4 z^3 a^3-3 z a^3+a^3 z^{-1} -z^5 a+z a+z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^6 z^{10}-a^4 z^{10}-3 a^7 z^9-7 a^5 z^9-4 a^3 z^9-2 a^8 z^8-5 a^6 z^8-11 a^4 z^8-8 a^2 z^8-a^9 z^7+11 a^7 z^7+17 a^5 z^7-5 a^3 z^7-10 a z^7+7 a^8 z^6+25 a^6 z^6+33 a^4 z^6+7 a^2 z^6-8 z^6+5 a^9 z^5-16 a^7 z^5-15 a^5 z^5+23 a^3 z^5+13 a z^5-4 z^5 a^{-1} -6 a^8 z^4-30 a^6 z^4-30 a^4 z^4+3 a^2 z^4-z^4 a^{-2} +8 z^4-8 a^9 z^3+16 a^7 z^3+18 a^5 z^3-14 a^3 z^3-6 a z^3+2 z^3 a^{-1} +14 a^6 z^2+15 a^4 z^2-a^2 z^2-2 z^2+4 a^9 z-10 a^7 z-14 a^5 z+a^3 z+a z-3 a^6-3 a^4-a^2+2 a^7 z^{-1} +3 a^5 z^{-1} +a^3 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 7 3     4 -2               9 6       -3 -4             8 6         2 -6           8 9           1 -8         5 8             -3 -10       4 8               4 -12     2 5                 -3 -14     4                   4 -16 1 2                     -1 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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