L11n192

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L11n191.gif

L11n191

L11n193.gif

L11n193

Contents

L11n192.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n192 at Knotilus!


Link Presentations

[edit Notes on L11n192's Link Presentations]

Planar diagram presentation X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X19,7,20,22 X11,20,12,21 X21,10,22,11 X2,14,3,13 X12,4,13,3
Gauss code {1, -10, 11, -3, 4, -6}, {3, -1, 2, 9, -8, -11, 10, -2, 5, -4, 6, -5, -7, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n192 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v-u-v+2) (2 u v-u-v+1)}{u v} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-9 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{7}{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 -a^3 z^3-z^3 a^{-3} -a^3 z-z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +2 a z^3+2 z^3 a^{-1} +a z+2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^3 a^{-5} -z a^{-5} +a^4 z^6-3 a^4 z^4+3 z^4 a^{-4} +2 a^4 z^2-2 z^2 a^{-4} +3 a^3 z^7+z^7 a^{-3} -10 a^3 z^5+8 a^3 z^3+2 z^3 a^{-3} -2 a^3 z- a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -8 a^2 z^6-5 z^6 a^{-2} +3 a^2 z^4+9 z^4 a^{-2} -4 z^2 a^{-2} + a^{-2} +a z^9+z^9 a^{-1} +3 a z^7+z^7 a^{-1} -16 a z^5-6 z^5 a^{-1} +13 a z^3+6 z^3 a^{-1} -3 a z- a^{-1} z^{-1} +5 z^8-14 z^6+12 z^4-4 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 \
-5-4-3-2-101234χ
10         11
8        2 -2
6       41 3
4      43  -1
2     53   2
0    45    1
-2   34     -1
-4  24      2
-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}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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L11n191.gif

L11n191

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L11n193