L9a52

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L9a51.gif

L9a51

L9a53.gif

L9a53

Contents

L9a52.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a52 at Knotilus!

L9a52 is 9^3_{8} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a52's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,11,17,12 X14,8,15,7 X8,14,9,13 X18,15,13,16 X12,17,5,18 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {5, -4, 6, -3, 7, -6}, {8, -1, 4, -5, 9, -2, 3, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a52 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) (w-1) \left(2 u w-u+w^2-2 w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-8} +3 q^{-7} -5 q^{-6} +8 q^{-5} -7 q^{-4} +9 q^{-3} -7 q^{-2} +q+5 q^{-1} -2 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^8+3 a^6 z^2+a^6 z^{-2} +3 a^6-2 a^4 z^4-4 a^4 z^2-2 a^4 z^{-2} -4 a^4-a^2 z^4+a^2 z^{-2} +a^2+z^2+1 (db)
Kauffman polynomial a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-7 a^8 z^4+5 a^8 z^2-2 a^8+3 a^7 z^7-3 a^7 z^5-5 a^7 z^3+3 a^7 z+a^6 z^8+7 a^6 z^6-22 a^6 z^4+20 a^6 z^2+a^6 z^{-2} -9 a^6+6 a^5 z^7-8 a^5 z^5-2 a^5 z^3+7 a^5 z-2 a^5 z^{-1} +a^4 z^8+7 a^4 z^6-19 a^4 z^4+20 a^4 z^2+2 a^4 z^{-2} -10 a^4+3 a^3 z^7-2 a^3 z^5-a^3 z^3+5 a^3 z-2 a^3 z^{-1} +3 a^2 z^6-3 a^2 z^4+3 a^2 z^2+a^2 z^{-2} -3 a^2+2 a z^5-2 a z^3+z^4-2 z^2+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 \
-7-6-5-4-3-2-1012χ
3         11
1        1 -1
-1       41 3
-3      53  -2
-5     42   2
-7    35    2
-9   54     1
-11  25      3
-13 13       -2
-15 2        2
-171         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
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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L9a51.gif

L9a51

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L9a53