L11n261

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L11n260.gif

L11n260

L11n262.gif

L11n262

Contents

L11n261.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n261 at Knotilus!


Link Presentations

[edit Notes on L11n261's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,20,9,19 X20,16,21,15 X16,22,17,21 X12,18,13,17 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 7, -8, 9, -5, 6, -7, 8, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n261 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w^3-u v w+u v-u w^3+2 u w^2-2 u w+u-v w^3+2 v w^2-2 v w+v-w^3+w^2-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -q^{10}+q^9-q^8-q^7+3 q^6-3 q^5+5 q^4-3 q^3+5 q^2-2 q+1 (db)
Signature 4 (db)
HOMFLY-PT polynomial - a^{-10} z^{-2} - a^{-10} +2 z^2 a^{-8} +3 a^{-8} z^{-2} +5 a^{-8} -2 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} -z^6 a^{-4} -4 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial z^8 a^{-4} +z^8 a^{-6} +z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-2} +z^6 a^{-4} -2 z^6 a^{-6} -8 z^6 a^{-8} -6 z^6 a^{-10} -6 z^5 a^{-3} -15 z^5 a^{-5} -20 z^5 a^{-7} -17 z^5 a^{-9} -6 z^5 a^{-11} -4 z^4 a^{-2} -14 z^4 a^{-4} -3 z^4 a^{-6} +15 z^4 a^{-8} +8 z^4 a^{-10} +z^3 a^{-3} +16 z^3 a^{-5} +42 z^3 a^{-7} +37 z^3 a^{-9} +10 z^3 a^{-11} +6 z^2 a^{-2} +12 z^2 a^{-4} +2 z^2 a^{-6} -7 z^2 a^{-8} -3 z^2 a^{-10} +4 z a^{-3} -10 z a^{-5} -34 z a^{-7} -27 z a^{-9} -7 z a^{-11} -4 a^{-2} -3 a^{-4} +4 a^{-6} +5 a^{-8} + a^{-10} -2 a^{-3} z^{-1} +2 a^{-5} z^{-1} +10 a^{-7} z^{-1} +8 a^{-9} z^{-1} +2 a^{-11} z^{-1} + a^{-2} z^{-2} + a^{-4} z^{-2} -2 a^{-6} z^{-2} -3 a^{-8} z^{-2} - a^{-10} 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 \
-2-10123456789χ
21           1-1
19            0
17         11 0
15       31   -2
13      1 1   2
11     451    0
9    211     2
7   141      2
5  42        2
3 14         3
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

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

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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L11n262