L11a521

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L11a520.gif

L11a520

L11a522.gif

L11a522

Contents

L11a521.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a521 at Knotilus!


Link Presentations

[edit Notes on L11a521's Link Presentations]

Planar diagram presentation X8192 X14,4,15,3 X10,14,11,13 X22,10,13,9 X20,17,21,18 X18,8,19,7 X12,20,7,19 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,1,21
Gauss code {1, -9, 2, -10, 8, -11}, {6, -1, 4, -3, 9, -7}, {3, -2, 10, -8, 5, -6, 7, -5, 11, -4}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11a521 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2 w^3-2 u^2 v^2 w^2-2 u^2 v w^3+4 u^2 v w^2-2 u^2 v w+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-2 u v^2 w+2 u v w^3-5 u v w^2+5 u v w-2 u v+2 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{u v w^{3/2}} (db)
Jones polynomial q^{10}-3 q^9+7 q^8-11 q^7+16 q^6-17 q^5+18 q^4-15 q^3+12 q^2-7 q+4- q^{-1} (db)
Signature 4 (db)
HOMFLY-PT polynomial z^4 a^{-8} +3 z^2 a^{-8} + a^{-8} z^{-2} +2 a^{-8} -2 z^6 a^{-6} -8 z^4 a^{-6} -10 z^2 a^{-6} -2 a^{-6} z^{-2} -6 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +9 z^4 a^{-4} +8 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -z^2 a^{-2} + a^{-2} (db)
Kauffman polynomial z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} - a^{-10} +8 z^7 a^{-9} -10 z^5 a^{-9} +5 z^3 a^{-9} +9 z^8 a^{-8} -17 z^6 a^{-8} +17 z^4 a^{-8} -12 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +6 z^9 a^{-7} -5 z^7 a^{-7} -12 z^5 a^{-7} +17 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +10 z^8 a^{-6} -43 z^6 a^{-6} +50 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +11 z^9 a^{-5} -30 z^7 a^{-5} +16 z^5 a^{-5} +7 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +5 z^8 a^{-4} -35 z^6 a^{-4} +41 z^4 a^{-4} -17 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +5 z^9 a^{-3} -16 z^7 a^{-3} +12 z^5 a^{-3} -z^3 a^{-3} +4 z^8 a^{-2} -15 z^6 a^{-2} +16 z^4 a^{-2} -4 z^2 a^{-2} - a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +2 z^3 a^{-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 \
-3-2-1012345678χ
21           11
19          31-2
17         4  4
15        73  -4
13       94   5
11      87    -1
9     109     1
7    710      3
5   58       -3
3  38        5
1 14         -3
-1 3          3
-31           -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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L11a520

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L11a522