L11a63

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L11a62.gif

L11a62

L11a64.gif

L11a64

Contents

L11a63.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a63 at Knotilus!


Link Presentations

[edit Notes on L11a63's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X22,16,5,15 X16,11,17,12 X20,18,21,17 X18,10,19,9 X10,20,11,19 X8,21,9,22 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 5, -2, 11, -3, 4, -5, 6, -7, 8, -6, 9, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a63 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{17/2}-4 q^{15/2}+9 q^{13/2}-13 q^{11/2}+17 q^{9/2}-20 q^{7/2}+18 q^{5/2}-16 q^{3/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -5 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +8 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-6 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -9 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +12 z^7 a^{-7} -16 z^5 a^{-7} +9 z^3 a^{-7} -3 z a^{-7} + a^{-7} z^{-1} +10 z^8 a^{-6} -8 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +5 z^9 a^{-5} +9 z^7 a^{-5} -39 z^5 a^{-5} +31 z^3 a^{-5} -12 z a^{-5} +3 a^{-5} z^{-1} +z^{10} a^{-4} +17 z^8 a^{-4} -46 z^6 a^{-4} +31 z^4 a^{-4} -7 z^2 a^{-4} +8 z^9 a^{-3} -8 z^7 a^{-3} -28 z^5 a^{-3} +41 z^3 a^{-3} -18 z a^{-3} +3 a^{-3} z^{-1} +z^{10} a^{-2} +10 z^8 a^{-2} -40 z^6 a^{-2} +41 z^4 a^{-2} -13 z^2 a^{-2} +2 a^{-2} +3 z^9 a^{-1} +a z^7-4 z^7 a^{-1} -4 a z^5-13 z^5 a^{-1} +6 a z^3+26 z^3 a^{-1} -4 a z-13 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^8-11 z^6+13 z^4-5 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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         61 -5
12        73  4
10       106   -4
8      107    3
6     810     2
4    810      -2
2   510       5
0  26        -4
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology

(db, data source)

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

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