L9a12

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L9a11.gif

L9a11

L9a13.gif

L9a13

Contents

L9a12.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a12 at Knotilus!

L9a12 is 9^2_{14} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a12's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,13,5,14 X14,7,15,8 X16,9,17,10 X8,15,9,16 X10,17,11,18 X2536 X4,11,1,12
Gauss code {1, -8, 2, -9}, {8, -1, 4, -6, 5, -7, 9, -2, 3, -4, 6, -5, 7, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a12 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 u v^4+2 u v^3-2 u v^2+2 u v-u-v^5+2 v^4-2 v^3+2 v^2-2 v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{4}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{11} (-z)-2 a^{11} z^{-1} +3 a^9 z^3+9 a^9 z+5 a^9 z^{-1} -2 a^7 z^5-8 a^7 z^3-9 a^7 z-3 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-a^5 z (db)
Kauffman polynomial -z^4 a^{14}+2 z^2 a^{14}-a^{14}-2 z^5 a^{13}+3 z^3 a^{13}-2 z^6 a^{12}+z^4 a^{12}+2 z^2 a^{12}-2 z^7 a^{11}+3 z^5 a^{11}-5 z^3 a^{11}+5 z a^{11}-2 a^{11} z^{-1} -z^8 a^{10}-z^6 a^{10}+6 z^4 a^{10}-11 z^2 a^{10}+5 a^{10}-5 z^7 a^9+17 z^5 a^9-26 z^3 a^9+16 z a^9-5 a^9 z^{-1} -z^8 a^8-z^6 a^8+9 z^4 a^8-12 z^2 a^8+5 a^8-3 z^7 a^7+11 z^5 a^7-15 z^3 a^7+10 z a^7-3 a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5 (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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        21-1
-8       2  2
-10      22  0
-12     52   3
-14    23    1
-16   34     -1
-18  12      1
-20 13       -2
-22 1        1
-241         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\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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L9a11.gif

L9a11

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L9a13