L11a215

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L11a214.gif

L11a214

L11a216.gif

L11a216

Contents

L11a215.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a215 at Knotilus!


Link Presentations

[edit Notes on L11a215's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X22,12,7,11 X20,16,21,15 X18,10,19,9 X10,20,11,19 X14,22,15,21 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17
Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a215 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-4 u^2 v^3+5 u^2 v^2-4 u^2 v+u^2-3 u v^4+6 u v^3-7 u v^2+6 u v-3 u+v^4-4 v^3+5 v^2-4 v+2}{u v^2} (db)
Jones polynomial -16 q^{9/2}+11 q^{7/2}-8 q^{5/2}+3 q^{3/2}+q^{23/2}-3 q^{21/2}+7 q^{19/2}-12 q^{17/2}+16 q^{15/2}-18 q^{13/2}+18 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +3 z^3 a^{-3} -z^3 a^{-5} -7 z^3 a^{-7} +3 z^3 a^{-9} +3 z a^{-3} +2 z a^{-5} -7 z a^{-7} +3 z a^{-9} + a^{-3} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-6} -2 z^{10} a^{-8} -4 z^9 a^{-5} -11 z^9 a^{-7} -7 z^9 a^{-9} -3 z^8 a^{-4} -4 z^8 a^{-6} -11 z^8 a^{-8} -10 z^8 a^{-10} -z^7 a^{-3} +10 z^7 a^{-5} +30 z^7 a^{-7} +10 z^7 a^{-9} -9 z^7 a^{-11} +10 z^6 a^{-4} +29 z^6 a^{-6} +45 z^6 a^{-8} +20 z^6 a^{-10} -6 z^6 a^{-12} +4 z^5 a^{-3} -z^5 a^{-5} -22 z^5 a^{-7} +14 z^5 a^{-11} -3 z^5 a^{-13} -9 z^4 a^{-4} -33 z^4 a^{-6} -50 z^4 a^{-8} -19 z^4 a^{-10} +6 z^4 a^{-12} -z^4 a^{-14} -6 z^3 a^{-3} -6 z^3 a^{-5} +11 z^3 a^{-7} -2 z^3 a^{-9} -11 z^3 a^{-11} +2 z^3 a^{-13} +z^2 a^{-4} +15 z^2 a^{-6} +27 z^2 a^{-8} +9 z^2 a^{-10} -3 z^2 a^{-12} +z^2 a^{-14} +4 z a^{-3} -7 z a^{-7} -z a^{-9} +2 z a^{-11} + a^{-4} -3 a^{-6} -5 a^{-8} -2 a^{-10} - a^{-3} z^{-1} +2 a^{-7} z^{-1} + a^{-9} 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
24           1-1
22          2 2
20         51 -4
18        72  5
16       95   -4
14      97    2
12     99     0
10    79      -2
8   510       5
6  36        -3
4 16         5
2 2          -2
01           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=4 i=6
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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