L9a14

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L9a13.gif

L9a13

L9a15.gif

L9a15

Contents

L9a14.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a14 at Knotilus!

L9a14 is 9^2_{13} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a14's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X16,10,17,9 X18,12,5,11 X8,16,9,15 X10,18,11,17 X2536 X4,14,1,13
Gauss code {1, -8, 2, -9}, {8, -1, 3, -6, 4, -7, 5, -2, 9, -3, 6, -4, 7, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L9a14 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-v+1\right) \left(v^2+v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+q^{3/2}+q^{19/2}-2 q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial z^5 a^{-7} +4 z^3 a^{-7} +4 z a^{-7} +2 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -12 z^3 a^{-5} -11 z a^{-5} -5 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +7 z a^{-3} +3 a^{-3} z^{-1} (db)
Kauffman polynomial -z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} +4 z^6 a^{-4} +z^6 a^{-6} -3 z^6 a^{-8} +6 z^5 a^{-3} +20 z^5 a^{-5} +11 z^5 a^{-7} -3 z^5 a^{-9} -z^4 a^{-4} +8 z^4 a^{-6} +6 z^4 a^{-8} -3 z^4 a^{-10} -12 z^3 a^{-3} -30 z^3 a^{-5} -12 z^3 a^{-7} +4 z^3 a^{-9} -2 z^3 a^{-11} -8 z^2 a^{-4} -12 z^2 a^{-6} +3 z^2 a^{-10} -z^2 a^{-12} +10 z a^{-3} +17 z a^{-5} +7 z a^{-7} +5 a^{-4} +5 a^{-6} - a^{-10} -3 a^{-3} z^{-1} -5 a^{-5} z^{-1} -2 a^{-7} 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 \
-2-101234567χ
20         1-1
18        1 1
16       21 -1
14      11  0
12     32   -1
10    11    0
8   13     2
6  21      1
4 13       2
2          0
01         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=4 i=6
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\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

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See/edit the Link Page master template (intermediate).

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L9a13.gif

L9a13

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L9a15