L7n1

L7a7

L7n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n1 at Knotilus!
	L7n1 is [math]\displaystyle{ 7^2_7 }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L7n1's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,10,6,11 X3849 X9,14,10,5 X11,2,12,3
Gauss code	{1, 7, -5, -3}, {-4, -1, 2, 5, -6, 4, -7, -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{-u v^3-1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} +a^7 z^3+4 a^7 z+3 a^7 z^{-1} -a^5 z^5-5 a^5 z^3-6 a^5 z-2 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10}+a^9 z-a^9 z^{-1} +a^8 z^4-4 a^8 z^2+3 a^8+a^7 z^5-5 a^7 z^3+7 a^7 z-3 a^7 z^{-1} +a^6 z^4-4 a^6 z^2+3 a^6+a^5 z^5-5 a^5 z^3+6 a^5 z-2 a^5 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   \ -5 -4 -3 -2 -1 0 χ -4           1 1 -6           1 1 -8       1     1 -10   1         1 -12   2 1       1 -14             0 -16 1           -1

Integral Khovanov Homology (db, data source)

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

L7a7 L7n2