L11n118

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L11n117.gif

L11n117

L11n119.gif

L11n119

Contents

L11n118.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n118 at Knotilus!


Link Presentations

[edit Notes on L11n118's Link Presentations]

Planar diagram presentation X6172 X3,15,4,14 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 X2536 X13,1,14,4
Gauss code {1, -10, -2, 11}, {10, -1, -6, 5, -3, 8, -4, 7, -11, 2, -8, 3, -9, 6, -7, 4, -5, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n118 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-2) (v-1) (2 v-1)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -10 q^{9/2}+11 q^{7/2}-13 q^{5/2}+11 q^{3/2}-\frac{2}{q^{3/2}}+q^{15/2}-3 q^{13/2}+7 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial 2 z^5 a^{-3} -4 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} +2 a z-6 z a^{-1} +8 z a^{-3} -5 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} (db)
Kauffman polynomial -2 z^9 a^{-3} -2 z^9 a^{-5} -6 z^8 a^{-2} -10 z^8 a^{-4} -4 z^8 a^{-6} -5 z^7 a^{-1} -6 z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} +17 z^6 a^{-2} +27 z^6 a^{-4} +8 z^6 a^{-6} -z^6 a^{-8} -z^6+13 z^5 a^{-1} +30 z^5 a^{-3} +25 z^5 a^{-5} +8 z^5 a^{-7} -24 z^4 a^{-2} -22 z^4 a^{-4} +z^4 a^{-6} +3 z^4 a^{-8} -4 z^4-3 a z^3-23 z^3 a^{-1} -39 z^3 a^{-3} -25 z^3 a^{-5} -6 z^3 a^{-7} +11 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2 a^{-8} +4 z^2+4 a z+13 z a^{-1} +18 z a^{-3} +12 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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 \
-2-101234567χ
16         1-1
14        2 2
12       51 -4
10      52  3
8     65   -1
6    75    2
4   46     2
2  57      -2
0 26       4
-2 3        -3
-42         2
Integral Khovanov Homology

(db, data source)

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

See/edit the Link_Splice_Base (expert).

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L11n117.gif

L11n117

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L11n119