L11n175

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L11n174.gif

L11n174

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L11n176

Contents

L11n175.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n175 at Knotilus!


Link Presentations

[edit Notes on L11n175's Link Presentations]

Planar diagram presentation X8192 X9,21,10,20 X14,5,15,6 X18,12,19,11 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code {1, 11, -5, -10, 3, -9}, {6, -1, -2, 5, 4, -6, 7, -3, 9, -7, 8, -4, -11, 2, 10, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n175 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-3 u^2 v^2+3 u^2 v-u^2+2 u v^4-5 u v^3+5 u v^2-5 u v+2 u-v^4+3 v^3-3 v^2}{u v^2} (db)
Jones polynomial -\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5+a^7 z^3-a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z+2 a^5 z^{-1} -2 a^3 z^3-3 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial a^{12} z^6-2 a^{12} z^4+4 a^{11} z^7-12 a^{11} z^5+7 a^{11} z^3+a^{11} z+5 a^{10} z^8-15 a^{10} z^6+11 a^{10} z^4-3 a^{10} z^2+2 a^9 z^9+2 a^9 z^7-19 a^9 z^5+17 a^9 z^3-5 a^9 z+a^9 z^{-1} +9 a^8 z^8-27 a^8 z^6+27 a^8 z^4-11 a^8 z^2+2 a^7 z^9+a^7 z^7-12 a^7 z^5+17 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-10 a^6 z^6+16 a^6 z^4-9 a^6 z^2+a^6+3 a^5 z^7-5 a^5 z^5+10 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6+2 a^4 z^4-a^4 z^2+3 a^3 z^3-4 a^3 z+a^3 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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        31-2
-6       51 4
-8      53  -2
-10     65   1
-12    55    0
-14   46     -2
-16  36      3
-18 13       -2
-20 3        3
-221         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}\oplus{\mathbb Z}_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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L11n174.gif

L11n174

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L11n176