L11n248

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L11n247.gif

L11n247

L11n249.gif

L11n249

Contents

L11n248.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n248 at Knotilus!


Link Presentations

[edit Notes on L11n248's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X15,11,16,22 X2,19,3,20 X21,17,22,16
Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, -9, 11, 8, -4, 10, -5, -11, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n248 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v+u^2-u v^2+3 u v-u+v^2-v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -6 q^{9/2}+10 q^{7/2}-\frac{1}{q^{7/2}}-14 q^{5/2}+\frac{4}{q^{5/2}}+15 q^{3/2}-\frac{9}{q^{3/2}}+2 q^{11/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-5} +z^5 a^{-3} +z^3 a^{-3} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +2 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial 3 z^4 a^{-6} -3 z^2 a^{-6} +z^7 a^{-5} +5 z^5 a^{-5} -7 z^3 a^{-5} +2 z a^{-5} +3 z^8 a^{-4} -z^6 a^{-4} +z^4 a^{-4} -z^2 a^{-4} +2 z^9 a^{-3} +4 z^7 a^{-3} +a^3 z^5-7 z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} +10 z^8 a^{-2} +4 a^2 z^6-14 z^6 a^{-2} -5 a^2 z^4+3 z^4 a^{-2} +a^2 z^2+2 z^2 a^{-2} +2 z^9 a^{-1} +8 a z^7+11 z^7 a^{-1} -15 a z^5-28 z^5 a^{-1} +9 a z^3+19 z^3 a^{-1} -4 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +7 z^8-9 z^6+z^2-1 (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 \
-4-3-2-1012345χ
12         2-2
10        4 4
8       62 -4
6      84  4
4     76   -1
2    88    0
0   69     3
-2  36      -3
-4 16       5
-6 3        -3
-81         1
Integral Khovanov Homology

(db, data source)

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

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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L11n247.gif

L11n247

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L11n249