L11n223

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L11n222.gif

L11n222

L11n224.gif

L11n224

Contents

L11n223.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n223 at Knotilus!


Link Presentations

[edit Notes on L11n223's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,8,19,7 X16,9,17,10 X22,15,9,16 X21,5,22,4 X5,14,6,15 X13,20,14,21 X8,18,1,17 X6,20,7,19
Gauss code {1, -2, 3, 7, -8, -11, 4, -10}, {5, -1, 2, -3, -9, 8, 6, -5, 10, -4, 11, 9, -7, -6}
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.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n223 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v+2 u v^2+2 u+v\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7+z^5 a^5+2 z^3 a^5-a^5 z^{-1} +2 z^5 a^3+7 z^3 a^3+7 z a^3+3 a^3 z^{-1} -2 z^3 a-5 z a-2 a z^{-1} (db)
Kauffman polynomial a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-5 a^8 z^4+3 a^8 z^2+4 a^7 z^7-11 a^7 z^5+15 a^7 z^3-4 a^7 z+3 a^6 z^8-8 a^6 z^6+12 a^6 z^4-5 a^6 z^2+a^6+a^5 z^9+a^5 z^7-6 a^5 z^5+7 a^5 z^3-a^5 z^{-1} +4 a^4 z^8-13 a^4 z^6+20 a^4 z^4-16 a^4 z^2+3 a^4+a^3 z^9-3 a^3 z^7+10 a^3 z^5-21 a^3 z^3+13 a^3 z-3 a^3 z^{-1} +a^2 z^8-2 a^2 z^6+2 a^2 z^4-6 a^2 z^2+3 a^2+3 a z^5-10 a z^3+9 a z-2 a 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 \
-7-6-5-4-3-2-1012χ
2         22
0        1 -1
-2       52 3
-4      43  -1
-6     43   1
-8    44    0
-10   34     -1
-12  14      3
-14 13       -2
-16 1        1
-181         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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L11n222.gif

L11n222

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L11n224