L8a7

L8a6

L8a8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a7 at Knotilus!
	L8a7 is [math]\displaystyle{ 8^2_{14} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a7's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,5,11,6 X12,3,13,4 X16,11,5,12 X2,9,3,10 X8,13,9,14
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}

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(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z-a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} -3 a^5 z^3-5 a^5 z-2 a^5 z^{-1} -a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-8 a^{10} z^4+5 a^{10} z^2+a^{10}+2 a^9 z^7-a^9 z^5-5 a^9 z^3+2 a^9 z-a^9 z^{-1} +8 a^8 z^6-16 a^8 z^4+4 a^8 z^2+3 a^8+2 a^7 z^7+4 a^7 z^5-12 a^7 z^3+6 a^7 z-3 a^7 z^{-1} +5 a^6 z^6-5 a^6 z^4-a^6 z^2+3 a^6+6 a^5 z^5-8 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +3 a^4 z^4+a^3 z^3 }[/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 χ -2                 1 1 -4               3 1 -2 -6             3     3 -8           2 3     1 -10         5 3       2 -12       2 3         1 -14     2 4           -2 -16   1 2             1 -18   2               -2 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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