Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L8a8

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L8a7.gif

L8a7

L8a9.gif

L8a9

Contents

L8a8.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a8's page at Knotilus.

Visit L8a8's page at the original Knot Atlas.

L8a8 is 8^2_{7} in the Rolfsen table of links, and the "seized Carrick bend" of practical knot-tying.


The simplest Celtic or pseudo-Celtic linear decorative knot.
Alternate decorative variant
Circular arcs only
Decorative variant with big loops at ends
Coat of arms of Bressauc, Jura, Switzerland.

Link Presentations

[edit Notes on L8a8's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X16,10,7,9 X2738 X14,12,15,11 X12,5,13,6 X4,13,5,14 X6,16,1,15
Gauss code {1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L8a8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1}{u v} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-4 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-3} +a^3 z-z a^{-3} +a^3 z^{-1} +z^5 a^{-1} -2 a z^3+3 z^3 a^{-1} -4 a z+3 z a^{-1} -a z^{-1} (db)
Kauffman polynomial -a z^7-z^7 a^{-1} -2 a^2 z^6-3 z^6 a^{-2} -5 z^6-a^3 z^5-2 a z^5-5 z^5 a^{-1} -4 z^5 a^{-3} +5 a^2 z^4+z^4 a^{-2} -3 z^4 a^{-4} +9 z^4+3 a^3 z^3+10 a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -2 a^2 z^2+2 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2-3 a^3 z-7 a z-6 z a^{-1} -2 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=1 is the signature of L8a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234χ
10        11
8       2 -2
6      21 1
4     32  -1
2    32   1
0   24    2
-2  22     0
-4 13      2
-6 1       -1
-81        1
Integral Khovanov Homology

(db, data source)

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

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.

L8a7.gif

L8a7

L8a9.gif

L8a9