L11a239

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L11a238

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L11a240

Contents

L11a239.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a239's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,10,21,9 X22,13,7,14 X14,21,15,22 X10,16,11,15 X18,5,19,6 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code {1, -9, 2, -10, 7, -11}, {9, -1, 3, -6, 10, -2, 4, -5, 6, -8, 11, -7, 8, -3, 5, -4}
A Braid Representative
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A Morse Link Presentation L11a239 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^3-6 u^2 v^2+7 u^2 v-2 u^2+u v^4-7 u v^3+13 u v^2-7 u v+u-2 v^4+7 v^3-6 v^2+2 v}{u v^2} (db)
Jones polynomial -\frac{17}{q^{9/2}}+\frac{20}{q^{7/2}}+q^{5/2}-\frac{20}{q^{5/2}}-4 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{12}{q^{11/2}}+8 \sqrt{q}-\frac{14}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} -4 a^7 z-2 a^7 z^{-1} +5 a^5 z^3+6 a^5 z+2 a^5 z^{-1} -2 a^3 z^5-3 a^3 z^3-4 a^3 z-a^3 z^{-1} -a z^5+z^3 a^{-1} -a z (db)
Kauffman polynomial a^9 z^7-4 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-10 a^8 z^6+10 a^8 z^4-3 a^8 z^2+4 a^7 z^9-9 a^7 z^7-2 a^7 z^5+12 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+6 a^6 z^8-35 a^6 z^6+39 a^6 z^4-15 a^6 z^2+a^6+12 a^5 z^9-28 a^5 z^7+6 a^5 z^5+15 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+16 a^4 z^8-57 a^4 z^6+53 a^4 z^4-17 a^4 z^2+8 a^3 z^9-6 a^3 z^7-16 a^3 z^5+17 a^3 z^3-5 a^3 z+a^3 z^{-1} +13 a^2 z^8-24 a^2 z^6+17 a^2 z^4+z^4 a^{-2} -5 a^2 z^2+12 a z^7-16 a z^5+4 z^5 a^{-1} +6 a z^3-2 z^3 a^{-1} +a z+8 z^6-6 z^4 (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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          3 3
2         51 -4
0        93  6
-2       106   -4
-4      108    2
-6     1010     0
-8    710      -3
-10   510       5
-12  37        -4
-14 16         5
-16 2          -2
-181           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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