L11a56

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L11a55.gif

L11a55

L11a57.gif

L11a57

Contents

L11a56.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a56 at Knotilus!


Link Presentations

[edit Notes on L11a56's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X16,10,17,9 X20,12,21,11 X22,18,5,17 X18,22,19,21 X8,16,9,15 X10,20,11,19 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 5, -2, 11, -3, 8, -4, 6, -7, 9, -5, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a56 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^4-2 t(2)^3+3 t(2)^2-2 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{23/2}-3 q^{21/2}+6 q^{19/2}-9 q^{17/2}+13 q^{15/2}-14 q^{13/2}+13 q^{11/2}-12 q^{9/2}+8 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial z^5 a^{-9} +3 z^3 a^{-9} +2 z a^{-9} + a^{-9} z^{-1} -z^7 a^{-7} -4 z^5 a^{-7} -5 z^3 a^{-7} -3 z a^{-7} - a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -5 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +5 z a^{-3} +2 a^{-3} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -6 z^9 a^{-7} -4 z^9 a^{-9} -2 z^8 a^{-4} -3 z^8 a^{-6} -7 z^8 a^{-8} -6 z^8 a^{-10} -z^7 a^{-3} +3 z^7 a^{-5} +15 z^7 a^{-7} +5 z^7 a^{-9} -6 z^7 a^{-11} +7 z^6 a^{-4} +17 z^6 a^{-6} +25 z^6 a^{-8} +10 z^6 a^{-10} -5 z^6 a^{-12} +5 z^5 a^{-3} +9 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +8 z^5 a^{-11} -3 z^5 a^{-13} -5 z^4 a^{-4} -15 z^4 a^{-6} -23 z^4 a^{-8} -6 z^4 a^{-10} +6 z^4 a^{-12} -z^4 a^{-14} -9 z^3 a^{-3} -16 z^3 a^{-5} -2 z^3 a^{-7} -3 z^3 a^{-9} -5 z^3 a^{-11} +3 z^3 a^{-13} -3 z^2 a^{-4} +4 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} -3 z^2 a^{-12} +z^2 a^{-14} +7 z a^{-3} +8 z a^{-5} +z a^{-11} +3 a^{-4} -3 a^{-8} + a^{-12} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + 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         41 -3
18        52  3
16       84   -4
14      65    1
12     78     1
10    56      -1
8   37       4
6  35        -2
4 15         4
2 1          -1
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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a55

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L11a57