L11a250

L11a249

L11a251

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Link Presentations

[edit Notes on L11a250's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,6,15,5 X22,18,9,17 X4,19,5,20 X6,22,7,21 X16,7,17,8 X8,9,1,10 X18,14,19,13 X20,15,21,16
Gauss code	{1, -2, 3, -6, 4, -7, 8, -9}, {9, -1, 2, -3, 10, -4, 11, -8, 5, -10, 6, -11, 7, -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{u^3 v^5-3 u^3 v^4+3 u^3 v^3-u^3 v^2-u^2 v^5+4 u^2 v^4-8 u^2 v^3+5 u^2 v^2-u^2 v-u v^4+5 u v^3-8 u v^2+4 u v-u-v^3+3 v^2-3 v+1}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{17}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^9-a^5 z^7+6 a^3 z^7-a z^7-4 a^5 z^5+11 a^3 z^5-4 a z^5-3 a^5 z^3+4 a^3 z^3-3 a z^3+2 a^5 z-5 a^3 z+a z+a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4+4 a^9 z^5-3 a^9 z^3+7 a^8 z^6-7 a^8 z^4+a^8 z^2+8 a^7 z^7-7 a^7 z^5-3 a^7 z^3+2 a^7 z+8 a^6 z^8-10 a^6 z^6-a^6 z^4+3 a^6 z^2+7 a^5 z^9-15 a^5 z^7+13 a^5 z^5-7 a^5 z^3-a^5 z+a^5 z^{-1} +3 a^4 z^{10}+a^4 z^8-21 a^4 z^6+21 a^4 z^4-2 a^4 z^2-a^4+13 a^3 z^9-48 a^3 z^7+54 a^3 z^5-17 a^3 z^3-4 a^3 z+a^3 z^{-1} +3 a^2 z^{10}-3 a^2 z^8-19 a^2 z^6+27 a^2 z^4-6 a^2 z^2+6 a z^9-24 a z^7+z^7 a^{-1} +27 a z^5-3 z^5 a^{-1} -9 a z^3+z^3 a^{-1} -a z+4 z^8-15 z^6+13 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     3   -3 2                   4 1   3 0                 6 3     -3 -2               9 4       5 -4             8 7         -1 -6           9 8           1 -8         6 8             2 -10       5 9               -4 -12     3 7                 4 -14   1 4                   -3 -16   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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