L11n332

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L11n331.gif

L11n331

L11n333.gif

L11n333

Contents

L11n332.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n332 at Knotilus!


Link Presentations

[edit Notes on L11n332's Link Presentations]

Planar diagram presentation X6172 X11,16,12,17 X3849 X17,2,18,3 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code {1, 4, -3, -11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, -4, -6, 11, 8, -10, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n332 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v+w-1) (v w-v-w) (u v w-1)}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial - q^{-9} +2 q^{-8} -3 q^{-7} +7 q^{-6} -5 q^{-5} +6 q^{-4} -5 q^{-3} +4 q^{-2} -2 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10}+z^4 a^8+4 z^2 a^8+a^8 z^{-2} +2 a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-2 a^6 z^{-2} -3 a^6-z^6 a^4-3 z^4 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2 (db)
Kauffman polynomial z^3 a^{11}-2 z a^{11}+2 z^4 a^{10}-4 z^2 a^{10}+3 a^{10}+z^7 a^9-4 z^5 a^9+10 z^3 a^9-7 z a^9+2 z^8 a^8-10 z^6 a^8+22 z^4 a^8-19 z^2 a^8-a^8 z^{-2} +11 a^8+z^9 a^7-2 z^7 a^7-3 z^5 a^7+14 z^3 a^7-12 z a^7+2 a^7 z^{-1} +4 z^8 a^6-17 z^6 a^6+27 z^4 a^6-22 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5-z^7 a^5-6 z^5 a^5+10 z^3 a^5-8 z a^5+2 a^5 z^{-1} +2 z^8 a^4-6 z^6 a^4+3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+2 z^7 a^3-7 z^5 a^3+5 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2 (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 \
-7-6-5-4-3-2-1012χ
1         11
-1        1 -1
-3       31 2
-5      32  -1
-7     32   1
-9   133    1
-11   53     2
-13  15      4
-15 12       -1
-17 1        1
-191         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{5} {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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L11n331

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L11n333