L11a421

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L11a420.gif

L11a420

L11a422.gif

L11a422

Contents

L11a421.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a421 at Knotilus!


Link Presentations

[edit Notes on L11a421's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,7,17,8 X20,9,21,10 X8,15,9,16 X10,19,5,20 X18,13,19,14 X22,17,11,18 X14,21,15,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -5, 4, -6}, {11, -2, 7, -9, 5, -3, 8, -7, 6, -4, 9, -8}
A Braid Representative
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A Morse Link Presentation L11a421 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{3 u v^2 w^2-3 u v^2 w+u v^2+3 u v w^3-7 u v w^2+5 u v w-u v+u w^4-4 u w^3+5 u w^2-2 u w+2 v^2 w^3-5 v^2 w^2+4 v^2 w-v^2+v w^4-5 v w^3+7 v w^2-3 v w-w^4+3 w^3-3 w^2}{\sqrt{u} v w^2} (db)
Jones polynomial  q^{-2} -4 q^{-3} +10 q^{-4} -15 q^{-5} +21 q^{-6} -22 q^{-7} +23 q^{-8} -18 q^{-9} +14 q^{-10} -8 q^{-11} +3 q^{-12} - q^{-13} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-9 z^2 a^{10}-5 a^{10} z^{-2} -13 a^{10}+5 z^4 a^8+9 z^2 a^8+2 a^8 z^{-2} +7 a^8+4 z^4 a^6+5 z^2 a^6+a^6+z^4 a^4 (db)
Kauffman polynomial z^7 a^{15}-4 z^5 a^{15}+6 z^3 a^{15}-4 z a^{15}+a^{15} z^{-1} +3 z^8 a^{14}-10 z^6 a^{14}+12 z^4 a^{14}-7 z^2 a^{14}-a^{14} z^{-2} +3 a^{14}+4 z^9 a^{13}-7 z^7 a^{13}-9 z^5 a^{13}+26 z^3 a^{13}-19 z a^{13}+5 a^{13} z^{-1} +2 z^{10} a^{12}+9 z^8 a^{12}-44 z^6 a^{12}+52 z^4 a^{12}-29 z^2 a^{12}-4 a^{12} z^{-2} +15 a^{12}+13 z^9 a^{11}-23 z^7 a^{11}-17 z^5 a^{11}+48 z^3 a^{11}-33 z a^{11}+9 a^{11} z^{-1} +2 z^{10} a^{10}+22 z^8 a^{10}-71 z^6 a^{10}+69 z^4 a^{10}-42 z^2 a^{10}-5 a^{10} z^{-2} +20 a^{10}+9 z^9 a^9-35 z^5 a^9+35 z^3 a^9-18 z a^9+5 a^9 z^{-1} +16 z^8 a^8-27 z^6 a^8+19 z^4 a^8-15 z^2 a^8-2 a^8 z^{-2} +8 a^8+15 z^7 a^7-19 z^5 a^7+7 z^3 a^7+10 z^6 a^6-9 z^4 a^6+5 z^2 a^6-a^6+4 z^5 a^5+z^4 a^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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-3           11
-5          41-3
-7         6  6
-9        94  -5
-11       126   6
-13      1110    -1
-15     1211     1
-17    813      5
-19   610       -4
-21  28        6
-23 16         -5
-25 2          2
-271           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-7 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-6 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=-5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a420

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L11a422