L9a51

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L9a50.gif

L9a50

L9a52.gif

L9a52

Contents

L9a51.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a51 at Knotilus!

L9a51 is 9^3_{11} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a51's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code {1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L9a51 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{\sqrt{u} v w} (db)
Jones polynomial q^7-3 q^6+7 q^5-8 q^4+11 q^3-10 q^2- q^{-2} +9 q+4 q^{-1} -6 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} -2 z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -3 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} + a^{-2} -z^4-z^2+1 (db)
Kauffman polynomial z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -10 z^4 a^{-6} +10 z^2 a^{-6} + a^{-6} z^{-2} -5 a^{-6} +5 z^7 a^{-5} -3 z^5 a^{-5} -4 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} +8 z^6 a^{-4} -21 z^4 a^{-4} +17 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +10 z^7 a^{-3} -16 z^5 a^{-3} +2 z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} +6 z^6 a^{-2} -18 z^4 a^{-2} +9 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +5 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-8 z^4+3 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 \
-3-2-10123456χ
15         11
13        31-2
11       4  4
9      43  -1
7     74   3
5    45    1
3   56     -1
1  36      3
-1 13       -2
-3 3        3
-51         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
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}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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L9a50.gif

L9a50

L9a52.gif

L9a52