L11a166

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L11a165.gif

L11a165

L11a167.gif

L11a167

Contents

L11a166.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a166 at Knotilus!


Link Presentations

[edit Notes on L11a166's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,11,21,12 X22,18,7,17 X18,22,19,21 X12,19,13,20
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -11, 5, -6, 7, -5, 9, -10, 11, -8, 10, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a166 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-5 u^2 v^3+5 u^2 v^2-u^2 v-u v^4+5 u v^3-7 u v^2+5 u v-u-v^3+5 v^2-5 v+1}{u v^2} (db)
Jones polynomial \frac{12}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+2 q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{10}{q^{11/2}}-5 \sqrt{q}+\frac{8}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-2 a^7 z+2 a^5 z^5+6 a^5 z^3+4 a^5 z+a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +2 a z^5+7 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -3 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+a^9 z+5 a^8 z^6-5 a^8 z^4+2 a^8 z^2+6 a^7 z^7-7 a^7 z^5+4 a^7 z^3-a^7 z+5 a^6 z^8-4 a^6 z^6-2 a^6 z^4+2 a^6 z^2+3 a^5 z^9+a^5 z^7-12 a^5 z^5+12 a^5 z^3-6 a^5 z+a^5 z^{-1} +a^4 z^{10}+5 a^4 z^8-15 a^4 z^6+8 a^4 z^4-a^4 z^2+5 a^3 z^9-10 a^3 z^7-4 a^3 z^5+15 a^3 z^3-10 a^3 z+2 a^3 z^{-1} +a^2 z^{10}+2 a^2 z^8-14 a^2 z^6+13 a^2 z^4-3 a^2 z^2+a^2+2 a z^9-4 a z^7+z^7 a^{-1} -7 a z^5-5 z^5 a^{-1} +18 a z^3+8 z^3 a^{-1} -11 a z+2 a z^{-1} -5 z a^{-1} + a^{-1} z^{-1} +2 z^8-8 z^6+9 z^4-3 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 \
-7-6-5-4-3-2-101234χ
6           11
4          1 -1
2         41 3
0        41  -3
-2       74   3
-4      75    -2
-6     76     1
-8    68      2
-10   46       -2
-12  26        4
-14 14         -3
-16 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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L11a165

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L11a167