L8a21

L8a20

L8n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a21 at Knotilus!
	L8a21 is a closed four-link chain. It is [math]\displaystyle{ 8^4_{1} }[/math] in the Rolfsen table of links.

Four linked squares.

Floor decoration in the Biblioteca Medicea Laurenziana, Florence, intended to contain four rings interlinked in this maner (but there is an interlacing error at upper right).

Ornamental circular arcs in square.

Link Presentations

[edit Notes on L8a21's Link Presentations]

Planar diagram presentation	X6172 X2536 X16,11,13,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X12,15,9,16
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -8}, {7, -6, 8, -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{t(2) t(1)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)-t(3) t(4) t(1)+2 t(4) t(1)-t(1)-t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-t(2) t(3) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^{-3} -3 a^9 z^{-3} -4 a^9 z^{-1} +3 a^7 z^{-3} +6 a^7 z+8 a^7 z^{-1} -3 a^5 z^3-a^5 z^{-3} -6 a^5 z-4 a^5 z^{-1} -a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+4 z^3 a^{11}-6 z a^{11}+4 a^{11} z^{-1} -a^{11} z^{-3} -z^6 a^{10}+6 z^2 a^{10}+3 a^{10} z^{-2} -8 a^{10}-z^7 a^9-2 z^5 a^9+11 z^3 a^9-14 z a^9+9 a^9 z^{-1} -3 a^9 z^{-3} -5 z^6 a^8+5 z^4 a^8+12 z^2 a^8+6 a^8 z^{-2} -15 a^8-z^7 a^7-7 z^5 a^7+17 z^3 a^7-14 z a^7+9 a^7 z^{-1} -3 a^7 z^{-3} -4 z^6 a^6+2 z^4 a^6+6 z^2 a^6+3 a^6 z^{-2} -8 a^6-6 z^5 a^5+9 z^3 a^5-6 z a^5+4 a^5 z^{-1} -a^5 z^{-3} -3 z^4 a^4-z^3 a^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           1 3     2 -10         6 3       3 -12       4 7         3 -14     1             1 -16     4             4 -18 1 1               0 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L8a20 L8n1