L9a24

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L9a23.gif

L9a23

L9a25.gif

L9a25

Contents

L9a24.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a24 at Knotilus!

L9a24 is 9^2_{21} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a24's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X18,13,7,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10
Gauss code {1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -4, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a24 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^3-u^2 v^2+u^2 v-u^2+u v^4-3 u v^3+3 u v^2-3 u v+u-v^4+v^3-v^2+v}{u v^2} (db)
Jones polynomial -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+3 z^3 a^5+4 z a^5+3 a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-2 a^3 z^{-1} -z^3 a-2 z a (db)
Kauffman polynomial a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-8 a^6 z^4+11 a^6 z^2-3 a^6+5 a^5 z^7-15 a^5 z^5+22 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8+a^4 z^6-6 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-4 a^3 z^5+3 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +2 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-3 a z^3+2 a z (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 \
-7-6-5-4-3-2-1012χ
2         11
0        1 -1
-2       31 2
-4      32  -1
-6     42   2
-8    34    1
-10   33     0
-12  13      2
-14 13       -2
-16 1        1
-181         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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L9a23.gif

L9a23

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L9a25