L11a36

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L11a35

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L11a37

Contents

L11a36.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a36 at Knotilus!


Link Presentations

[edit Notes on L11a36's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,15,19,16 X16,7,17,8 X8,17,9,18 X20,11,21,12 X22,13,5,14 X12,21,13,22 X14,19,15,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -8, 7, -9, 3, -4, 5, -3, 9, -6, 8, -7}
A Braid Representative
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A Morse Link Presentation L11a36 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 u v^4-6 u v^3+7 u v^2-5 u v+2 u+2 v^5-5 v^4+7 v^3-6 v^2+3 v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{3}{q^{25/2}}+\frac{7}{q^{23/2}}-\frac{10}{q^{21/2}}+\frac{13}{q^{19/2}}-\frac{15}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{13}{q^{13/2}}+\frac{8}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^{13}-a^{13} z^{-1} +3 z^3 a^{11}+5 z a^{11}+a^{11} z^{-1} -2 z^5 a^9-4 z^3 a^9+2 a^9 z^{-1} -2 z^5 a^7-5 z^3 a^7-4 z a^7-2 a^7 z^{-1} -z^5 a^5-3 z^3 a^5-2 z a^5 (db)
Kauffman polynomial a^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3-a^{15} z+4 a^{14} z^8-8 a^{14} z^6+a^{14} z^2+3 a^{13} z^9-2 a^{13} z^7-9 a^{13} z^5+6 a^{13} z^3-a^{13} z^{-1} +a^{12} z^{10}+6 a^{12} z^8-21 a^{12} z^6+22 a^{12} z^4-12 a^{12} z^2+3 a^{12}+6 a^{11} z^9-13 a^{11} z^7+10 a^{11} z^5-a^{11} z^3+a^{11} z-a^{11} z^{-1} +a^{10} z^{10}+5 a^{10} z^8-17 a^{10} z^6+21 a^{10} z^4-5 a^{10} z^2+3 a^9 z^9-5 a^9 z^7+4 a^9 z^5+7 a^9 z^3-8 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-3 a^8 z^6-2 a^8 z^4+6 a^8 z^2-3 a^8+3 a^7 z^7-6 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-3 a^5 z^3+2 a^5 z (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          21-1
-8         3  3
-10        52  -3
-12       83   5
-14      76    -1
-16     87     1
-18    57      2
-20   58       -3
-22  25        3
-24 15         -4
-26 2          2
-281           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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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L11a35

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L11a37