L9a1

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L8n8.gif

L8n8

L9a2.gif

L9a2

Contents

L9a1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a1 at Knotilus!

L9a1 is 9^2_{32} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,12,5,11 X12,18,13,17 X16,10,17,9 X2,14,3,13
Gauss code {1, -9, 5, -3}, {4, -1, 2, -5, 8, -4, 6, -7, 9, -2, 3, -8, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-10 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{1}{q^{5/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} -a z^3+z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} -z a^{-1} +2 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -5 z^6 a^{-2} -2 z^6 a^{-4} -3 z^6 a^{-6} -6 z^6-4 a z^5+4 z^5 a^{-1} +18 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+11 z^4 a^{-2} +10 z^4 a^{-4} +7 z^4 a^{-6} +7 z^4+4 a z^3-13 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -4 z^2 a^{-2} -7 z^2 a^{-4} -4 z^2 a^{-6} -z^2+2 z a^{-1} +4 z a^{-3} +2 z a^{-5} +1-a z^{-1} - a^{-1} 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 \
-3-2-10123456χ
14         11
12        2 -2
10       31 2
8      52  -3
6     53   2
4    45    1
2   55     0
0  36      3
-2 13       -2
-4 3        3
-61         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L8n8.gif

L8n8

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L9a2