L11n417

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L11n416.gif

L11n416

L11n418.gif

L11n418

Contents

L11n417.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n417 at Knotilus!


Link Presentations

[edit Notes on L11n417's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X15,21,16,20 X5,15,6,14 X13,5,14,4 X19,7,20,12 X11,19,12,18 X17,13,18,22 X21,17,22,16 X2738 X6,9,1,10
Gauss code {1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n417 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v w^3-u^2 w^3+u^2 w^2-u^2 w+u^2+u v^2 w^3-u v w^3+u v-u-v^2 w^3+v^2 w^2-v^2 w+v^2-v}{u v w^{3/2}} (db)
Jones polynomial -q^7+2 q^6-3 q^5+4 q^4-4 q^3+5 q^2-3 q+4- q^{-1} + q^{-2} (db)
Signature 4 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-9 z^2 a^{-2} -2 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-10 a^{-2} +4 a^{-4} +2 a^{-6} - a^{-8} +5-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} (db)
Kauffman polynomial z a^{-9} +2 z^2 a^{-8} - a^{-8} +z^5 a^{-7} -z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +3 z^6 a^{-6} -10 z^4 a^{-6} +6 z^2 a^{-6} - a^{-6} z^{-2} +5 z^7 a^{-5} -23 z^5 a^{-5} +31 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +4 z^8 a^{-4} -19 z^6 a^{-4} +27 z^4 a^{-4} -21 z^2 a^{-4} -4 a^{-4} z^{-2} +13 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -25 z^5 a^{-3} +50 z^3 a^{-3} -35 z a^{-3} +9 a^{-3} z^{-1} +5 z^8 a^{-2} -29 z^6 a^{-2} +55 z^4 a^{-2} -46 z^2 a^{-2} -5 a^{-2} z^{-2} +22 a^{-2} +z^9 a^{-1} -4 z^7 a^{-1} -z^5 a^{-1} +18 z^3 a^{-1} -19 z a^{-1} +5 a^{-1} z^{-1} +z^8-7 z^6+18 z^4-21 z^2-2 z^{-2} +11 (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 \
-4-3-2-1012345χ
15         1-1
13        1 1
11       32 -1
9      221 1
7     33   0
5    221   1
3   24     2
1  21      1
-1 14       3
-3          0
-51         1
Integral Khovanov Homology

(db, data source)

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

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

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L11n418