L7a3

L7a2

L7a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a3 at Knotilus!
	L7a3 is [math]\displaystyle{ 7^2_4 }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a3's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X14,10,5,9 X8,14,9,13 X2536 X4,12,1,11
Gauss code	{1, -6, 2, -7}, {6, -1, 3, -5, 4, -2, 7, -3, 5, -4}

A Braid Representative	{{{braid_table}}}
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(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+q^{13/2}-2 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -5 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-8} +2 z^3 a^{-7} +3 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +3 z^5 a^{-5} -6 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +z^6 a^{-4} +z^4 a^{-4} -6 z^2 a^{-4} +3 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +9 z a^{-3} -3 a^{-3} z^{-1} +z^6 a^{-2} -2 z^4 a^{-2} -2 z^2 a^{-2} +3 a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +5 z a^{-1} -2 a^{-1} 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   \ -2 -1 0 1 2 3 4 5 χ 14               1 -1 12             1   1 10           2 1   -1 8         1 1     0 6       1 2       1 4     2 1         1 2   1 3           2 0                 0 -2 1               1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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 }[/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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