L11a65

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L11a64.gif

L11a64

L11a66.gif

L11a66

Contents

L11a65.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a65's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X20,12,21,11 X22,16,5,15 X16,22,17,21 X8,20,9,19 X18,10,19,9 X10,18,11,17 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -7, 8, -9, 4, -2, 11, -3, 5, -6, 9, -8, 7, -4, 6, -5}
A Braid Representative
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A Morse Link Presentation L11a65 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(4 t(2)^2-5 t(2)+4\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial 16 q^{9/2}-15 q^{7/2}+10 q^{5/2}-7 q^{3/2}+q^{21/2}-3 q^{19/2}+7 q^{17/2}-10 q^{15/2}+14 q^{13/2}-17 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} +z a^{-9} + a^{-9} z^{-1} -z^5 a^{-7} -z^3 a^{-7} -z a^{-7} - a^{-7} z^{-1} -2 z^5 a^{-5} -4 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} +3 z a^{-3} +2 a^{-3} z^{-1} +z^3 a^{-1} +z a^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -3 z^9 a^{-5} -7 z^9 a^{-7} -4 z^9 a^{-9} -5 z^8 a^{-4} -9 z^8 a^{-6} -9 z^8 a^{-8} -5 z^8 a^{-10} -5 z^7 a^{-3} -4 z^7 a^{-5} +12 z^7 a^{-7} +8 z^7 a^{-9} -3 z^7 a^{-11} -3 z^6 a^{-2} +6 z^6 a^{-4} +27 z^6 a^{-6} +34 z^6 a^{-8} +15 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +9 z^5 a^{-3} +22 z^5 a^{-5} -4 z^5 a^{-9} +8 z^5 a^{-11} +5 z^4 a^{-2} -32 z^4 a^{-6} -46 z^4 a^{-8} -16 z^4 a^{-10} +3 z^4 a^{-12} +2 z^3 a^{-1} -7 z^3 a^{-3} -29 z^3 a^{-5} -14 z^3 a^{-7} +2 z^3 a^{-9} -4 z^3 a^{-11} -z^2 a^{-2} -3 z^2 a^{-4} +15 z^2 a^{-6} +31 z^2 a^{-8} +12 z^2 a^{-10} -2 z^2 a^{-12} -z a^{-1} +7 z a^{-3} +13 z a^{-5} +5 z a^{-7} +2 a^{-4} -4 a^{-6} -9 a^{-8} -4 a^{-10} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-7} z^{-1} + a^{-9} z^{-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 \
-2-10123456789χ
22           1-1
20          2 2
18         51 -4
16        52  3
14       95   -4
12      85    3
10     89     1
8    78      -1
6   38       5
4  47        -3
2 15         4
0 2          -2
-21           1
Integral Khovanov Homology

(db, data source)

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