L11n81

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L11n80.gif

L11n80

L11n82.gif

L11n82

Contents

L11n81.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n81 at Knotilus!


Link Presentations

[edit Notes on L11n81's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X5,14,6,15 X10,4,11,3 X11,16,12,17 X15,12,16,13 X13,22,14,5 X2,18,3,17
Gauss code {1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, 4, -3, 5, 10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n81 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{8}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{8}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^9-a^9 z^{-1} -z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+5 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-5 z a^3-2 a^3 z^{-1} (db)
Kauffman polynomial -3 z^2 a^{12}+a^{12}-z^5 a^{11}-3 z^3 a^{11}+z a^{11}-3 z^6 a^{10}+3 z^4 a^{10}-2 z^2 a^{10}-4 z^7 a^9+8 z^5 a^9-6 z^3 a^9+2 z a^9+a^9 z^{-1} -3 z^8 a^8+5 z^6 a^8-z^4 a^8+6 z^2 a^8-3 a^8-z^9 a^7-4 z^7 a^7+21 z^5 a^7-19 z^3 a^7+4 z a^7+a^7 z^{-1} -5 z^8 a^6+16 z^6 a^6-11 z^4 a^6+2 z^2 a^6-z^9 a^5-z^7 a^5+17 z^5 a^5-25 z^3 a^5+10 z a^5-2 a^5 z^{-1} -2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-3 z^2 a^4+3 a^4-z^7 a^3+5 z^5 a^3-9 z^3 a^3+7 z a^3-2 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 \
-7-6-5-4-3-2-1012χ
0         11
-2        1 -1
-4       41 3
-6      33  0
-8     52   3
-10    33    0
-12   45     -1
-14  23      1
-16 24       -2
-18 2        2
-202         -2
Integral Khovanov Homology

(db, data source)

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