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 at Knotilus!

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).   
\ 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).

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L8a7

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L8a9