L7a2

L7a1

L7a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a2 at Knotilus!
	L7a2 is [math]\displaystyle{ 7^2_5 }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a2's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,11,5,12 X12,7,13,8 X8,13,9,14 X2536 X4,9,1,10
Gauss code	{1, -6, 2, -7}, {6, -1, 4, -5, 7, -2, 3, -4, 5, -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{-2 u v^2+2 u v-u-v^3+2 v^2-2 v}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} +3 a^7 z+3 a^7 z^{-1} -2 a^5 z^3-4 a^5 z-2 a^5 z^{-1} -a^3 z^3-a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+a^{10}+2 a^9 z^5-4 a^9 z^3+2 a^9 z-a^9 z^{-1} +a^8 z^6+2 a^8 z^4-7 a^8 z^2+3 a^8+5 a^7 z^5-10 a^7 z^3+8 a^7 z-3 a^7 z^{-1} +a^6 z^6+3 a^6 z^4-6 a^6 z^2+3 a^6+3 a^5 z^5-5 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +2 a^4 z^4-a^4 z^2+a^3 z^3-a^3 z }[/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 χ -2               1 1 -4             2 1 -1 -6           2     2 -8         1 2     1 -10       3 2       1 -12     1 2         1 -14   1 2           -1 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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