L11a64

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L11a63.gif

L11a63

L11a65.gif

L11a65

Contents

L11a64.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a64 at Knotilus!


Link Presentations

[edit Notes on L11a64's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X18,10,19,9 X22,16,5,15 X16,22,17,21 X20,12,21,11 X8,18,9,17 X10,20,11,19 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 7, -2, 11, -3, 5, -6, 8, -4, 9, -7, 6, -5}
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.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a64 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 -12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}+q^{23/2}-3 q^{21/2}+7 q^{19/2}-10 q^{17/2}+13 q^{15/2}-14 q^{13/2}+13 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -4 z^3 a^{-5} -6 z^3 a^{-7} +3 z^3 a^{-9} +4 z a^{-3} -7 z a^{-7} +3 z a^{-9} + a^{-3} z^{-1} + a^{-5} z^{-1} -4 a^{-7} z^{-1} +2 a^{-9} z^{-1} (db)
Kauffman polynomial z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -2 z^3 a^{-13} +6 z^6 a^{-12} -8 z^4 a^{-12} +6 z^2 a^{-12} -2 a^{-12} +7 z^7 a^{-11} -9 z^5 a^{-11} +4 z^3 a^{-11} +z a^{-11} +6 z^8 a^{-10} -7 z^6 a^{-10} +4 z^2 a^{-10} - a^{-10} +4 z^9 a^{-9} -6 z^7 a^{-9} +9 z^5 a^{-9} -15 z^3 a^{-9} +8 z a^{-9} -2 a^{-9} z^{-1} +z^{10} a^{-8} +6 z^8 a^{-8} -21 z^6 a^{-8} +24 z^4 a^{-8} -19 z^2 a^{-8} +6 a^{-8} +6 z^9 a^{-7} -19 z^7 a^{-7} +25 z^5 a^{-7} -24 z^3 a^{-7} +13 z a^{-7} -4 a^{-7} z^{-1} +z^{10} a^{-6} +2 z^8 a^{-6} -16 z^6 a^{-6} +23 z^4 a^{-6} -16 z^2 a^{-6} +5 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -z^5 a^{-5} +5 z^3 a^{-5} +z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + a^{-3} 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        52  3
16       85   -3
14      65    1
12     78     1
10    56      -1
8   27       5
6  35        -2
4 14         3
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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a63

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L11a65