L10a18

L10a17

L10a19

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Link Presentations

[edit Notes on L10a18's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X18,13,19,14 X16,7,17,8 X8,17,9,18 X20,15,5,16 X14,19,15,20 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7}

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-1) (v-1)^3 \left(v^2+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{10}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{9}{q^{5/2}}+q^{3/2}-\frac{9}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-3 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+8 a^5 z^3+11 a^5 z+5 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-10 a^3 z^3-10 a^3 z-3 a^3 z^{-1} +a z^5+3 a z^3+2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+2 z^2 a^{10}-a^{10}-2 z^5 a^9+2 z^3 a^9-3 z^6 a^8+2 z^4 a^8-4 z^7 a^7+6 z^5 a^7-8 z^3 a^7+5 z a^7-2 a^7 z^{-1} -3 z^8 a^6+z^6 a^6+6 z^4 a^6-11 z^2 a^6+5 a^6-z^9 a^5-7 z^7 a^5+24 z^5 a^5-29 z^3 a^5+17 z a^5-5 a^5 z^{-1} -6 z^8 a^4+11 z^6 a^4+2 z^4 a^4-11 z^2 a^4+5 a^4-z^9 a^3-6 z^7 a^3+26 z^5 a^3-29 z^3 a^3+16 z a^3-3 a^3 z^{-1} -3 z^8 a^2+6 z^6 a^2+2 z^4 a^2-4 z^2 a^2-3 z^7 a+10 z^5 a-10 z^3 a+4 z a-z^6+3 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 χ 4                     1 -1 2                   2   2 0                 3 1   -2 -2               6 2     4 -4             5 5       0 -6           6 4         2 -8         4 5           1 -10       4 6             -2 -12     1 4               3 -14   1 4                 -3 -16   1                   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=-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}\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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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