L11a360

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L11a359.gif

L11a359

L11a361.gif

L11a361

Contents

L11a360.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a360 at Knotilus!


Link Presentations

[edit Notes on L11a360's Link Presentations]

Planar diagram presentation X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X18,8,19,7 X20,10,21,9 X2,11,3,12 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,1,21
Gauss code {1, -7, 2, -8, 4, -9, 5, -10, 6, -11}, {7, -1, 3, -2, 8, -4, 9, -5, 10, -6, 11, -3}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11a360 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^4 v^4-u^4 v^3-u^3 v^4+u^3 v^3-u^3 v^2-u^2 v^3+u^2 v^2-u^2 v-u v^2+u v-u-v+1}{u^2 v^2} (db)
Jones polynomial 2 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}+q^{25/2}-2 q^{23/2}+2 q^{21/2}-3 q^{19/2}+3 q^{17/2}-3 q^{15/2}+3 q^{13/2}-3 q^{11/2} (db)
Signature 7 (db)
HOMFLY-PT polynomial -z^9 a^{-7} +z^7 a^{-5} -8 z^7 a^{-7} +z^7 a^{-9} +7 z^5 a^{-5} -22 z^5 a^{-7} +6 z^5 a^{-9} +15 z^3 a^{-5} -25 z^3 a^{-7} +10 z^3 a^{-9} +10 z a^{-5} -11 z a^{-7} +4 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -z^9 a^{-5} -3 z^9 a^{-7} -2 z^9 a^{-9} +7 z^8 a^{-6} +5 z^8 a^{-8} -2 z^8 a^{-10} +8 z^7 a^{-5} +21 z^7 a^{-7} +11 z^7 a^{-9} -2 z^7 a^{-11} -15 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -2 z^6 a^{-12} -22 z^5 a^{-5} -48 z^5 a^{-7} -18 z^5 a^{-9} +6 z^5 a^{-11} -2 z^5 a^{-13} +9 z^4 a^{-6} -3 z^4 a^{-8} -6 z^4 a^{-10} +4 z^4 a^{-12} -2 z^4 a^{-14} +25 z^3 a^{-5} +42 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} +2 z^3 a^{-13} -2 z^3 a^{-15} +2 z^2 a^{-6} +3 z^2 a^{-8} -z^2 a^{-10} +z^2 a^{-14} -z^2 a^{-16} -11 z a^{-5} -13 z a^{-7} -3 z a^{-9} -z a^{-11} +z a^{-13} +z a^{-15} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-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 \
-2-10123456789χ
26           1-1
24          1 1
22         11 0
20        21  1
18       22   0
16      11    0
14     22     0
12    11      0
10   12       1
8  11        0
6 12         1
4            0
21           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=6 i=8
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\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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L11a359

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L11a361