L11a51

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L11a50.gif

L11a50

L11a52.gif

L11a52

Contents

L11a51.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a51 at Knotilus!


Link Presentations

[edit Notes on L11a51's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,6,15,5 X10,4,11,3 X20,12,21,11 X22,14,5,13 X12,22,13,21 X2,16,3,15
Gauss code {1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -10, 9, -6, 11, -2, 4, -3, 5, -8, 10, -9}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11a51 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2+1\right) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{7/2}} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-11 q^{5/2}+12 q^{3/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^9 a^{-1} -a z^7+7 z^7 a^{-1} -z^7 a^{-3} -5 a z^5+17 z^5 a^{-1} -5 z^5 a^{-3} -7 a z^3+16 z^3 a^{-1} -7 z^3 a^{-3} -2 a z+4 z a^{-1} -2 z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-4 a z^9-8 z^9 a^{-1} -4 z^9 a^{-3} -4 a^2 z^8+4 z^8 a^{-2} -4 z^8 a^{-4} +4 z^8-3 a^3 z^7+15 a z^7+36 z^7 a^{-1} +14 z^7 a^{-3} -4 z^7 a^{-5} -a^4 z^6+13 a^2 z^6+9 z^6 a^{-4} -3 z^6 a^{-6} +2 z^6+10 a^3 z^5-21 a z^5-65 z^5 a^{-1} -24 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-9 a^2 z^4-11 z^4 a^{-2} -6 z^4 a^{-4} +7 z^4 a^{-6} -10 z^4-6 a^3 z^3+16 a z^3+44 z^3 a^{-1} +16 z^3 a^{-3} -4 z^3 a^{-5} +2 z^3 a^{-7} -a^4 z^2+a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +5 z^2-4 a z-8 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} 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 \
-5-4-3-2-10123456χ
14           11
12          2 -2
10         31 2
8        52  -3
6       63   3
4      65    -1
2     76     1
0    58      3
-2   35       -2
-4  25        3
-6 13         -2
-8 2          2
-101           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L11a50

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L11a52