L11n359

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L11n358.gif

L11n358

L11n360.gif

L11n360

Contents

L11n359.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n359 at Knotilus!


Link Presentations

[edit Notes on L11n359's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X15,5,16,14 X19,15,20,22 X13,20,14,21 X21,12,22,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-6, 5, -4, 3, -7, 8, -9, 7}, {10, -1, -5, 4, 11, -2, -3, 9, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n359 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-t(2)^2 t(3)-t(1) t(2) t(3)+t(2) t(3)+t(1)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^5+ q^{-5} +2 q^4- q^{-4} -2 q^3+2 q^{-3} +3 q^2-2 q+2 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^6-2 a^2 z^4+6 z^4+a^4 z^2-9 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+3 a^4-8 a^2- a^{-2} +6+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-6 a^3 z^7-6 a z^7+z^7 a^{-1} +z^7 a^{-3} -7 a^4 z^6-22 a^2 z^6+2 z^6 a^{-2} +2 z^6 a^{-4} -15 z^6+8 a^3 z^5+5 a z^5-6 z^5 a^{-1} -2 z^5 a^{-3} +z^5 a^{-5} +16 a^4 z^4+48 a^2 z^4-7 z^4 a^{-2} -7 z^4 a^{-4} +32 z^4+2 a^3 z^3+10 a z^3+9 z^3 a^{-1} -2 z^3 a^{-3} -3 z^3 a^{-5} -16 a^4 z^2-40 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -24 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-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 \
-6-5-4-3-2-1012345χ
11           1-1
9          1 1
7         11 0
5       131  1
3      122   1
1     132    0
-1    123     2
-3   121      0
-5  112       2
-7 12         1
-9            0
-111           1
Integral Khovanov Homology

(db, data source)

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