L9n21

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L9n20.gif

L9n20

L9n22.gif

L9n22

Contents

L9n21.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n21 at Knotilus!

L9n21 is 9^3_{17} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n21's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X16,11,17,12 X18,15,9,16 X12,17,13,18 X2536 X9,1,10,4
Gauss code {1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, 5, -7, -4, 3, 6, -5, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n21 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v-u w^2+u w-v w^2+v w+w^3}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-5} + q^{-4} +q^3- q^{-3} + q^{-2} +q+2 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^4-a^4 z^{-2} -2 a^4+z^4 a^2+5 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^4-6 z^2-5 z^{-2} -9+z^2 a^{-2} +2 a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial a^4 z^6+a^2 z^6+z^6 a^{-2} +z^6+a^5 z^5+2 a^3 z^5+2 a z^5+z^5 a^{-1} -4 a^4 z^4-6 a^2 z^4-6 z^4 a^{-2} -8 z^4-4 a^5 z^3-10 a^3 z^3-13 a z^3-7 z^3 a^{-1} +3 a^4 z^2+11 a^2 z^2+10 z^2 a^{-2} +18 z^2+3 a^5 z+13 a^3 z+21 a z+11 z a^{-1} -2 a^4-10 a^2-7 a^{-2} -14-a^5 z^{-1} -5 a^3 z^{-1} -9 a z^{-1} -5 a^{-1} z^{-1} +a^4 z^{-2} +4 a^2 z^{-2} +2 a^{-2} z^{-2} +5 z^{-2} (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 \
-5-4-3-2-101234χ
7         11
5         11
3       1  1
1     3    3
-1    141   2
-3   1      1
-5   11     0
-7 11       0
-9          0
-111         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}^{3}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\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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L9n20

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L9n22