L11a357

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L11a356.gif

L11a356

L11a358.gif

L11a358

Contents

L11a357.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a357's Link Presentations]

Planar diagram presentation X12,1,13,2 X18,7,19,8 X14,3,15,4 X20,10,21,9 X22,19,11,20 X6,15,7,16 X16,5,17,6 X4,17,5,18 X8,22,9,21 X2,11,3,12 X10,13,1,14
Gauss code {1, -10, 3, -8, 7, -6, 2, -9, 4, -11}, {10, -1, 11, -3, 6, -7, 8, -2, 5, -4, 9, -5}
A Braid Representative
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A Morse Link Presentation L11a357 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u+v-1) (u v-u-v) \left(u^2 v+u v^2-u v+u+v\right)}{u^2 v^2} (db)
Jones polynomial \frac{15}{q^{9/2}}-\frac{13}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{14}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)-2 a^9 z+a^7 z^5+2 a^7 z^3+2 a^7 z+a^7 z^{-1} +2 a^5 z^5+4 a^5 z^3+a^5 z-a^5 z^{-1} +a^3 z^5-3 a^3 z-a z^3-a z (db)
Kauffman polynomial a^{12} z^6-4 a^{12} z^4+4 a^{12} z^2+2 a^{11} z^7-7 a^{11} z^5+8 a^{11} z^3-4 a^{11} z+2 a^{10} z^8-3 a^{10} z^6-3 a^{10} z^4+4 a^{10} z^2+2 a^9 z^9-3 a^9 z^7+a^9 z^5+2 a^9 z^3-6 a^9 z+a^8 z^{10}+a^8 z^8-a^8 z^6-4 a^8 z^4+4 a^8 z^2+5 a^7 z^9-11 a^7 z^7+15 a^7 z^5-11 a^7 z^3+4 a^7 z-a^7 z^{-1} +a^6 z^{10}+4 a^6 z^8-8 a^6 z^6+5 a^6 z^4+a^6 z^2+a^6+3 a^5 z^9-a^5 z^7-3 a^5 z^5+2 a^5 z^3+2 a^5 z-a^5 z^{-1} +5 a^4 z^8-8 a^4 z^6+5 a^4 z^4-2 a^4 z^2+5 a^3 z^7-9 a^3 z^5+5 a^3 z^3-3 a^3 z+3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          2 -2
-2         51 4
-4        63  -3
-6       74   3
-8      86    -2
-10     67     -1
-12    58      3
-14   46       -2
-16  16        5
-18 13         -2
-20 1          1
-221           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a356.gif

L11a356

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L11a358