L11a61

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L11a60.gif

L11a60

L11a62.gif

L11a62

Contents

L11a61.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a61 at Knotilus!


Link Presentations

[edit Notes on L11a61's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X22,13,5,14 X14,7,15,8 X18,9,19,10 X20,17,21,18 X16,21,17,22 X8,15,9,16 X10,19,11,20 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -8, 5, -9, 11, -2, 3, -4, 8, -7, 6, -5, 9, -6, 7, -3}
A Braid Representative
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A Morse Link Presentation L11a61 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^5+4 t(1) t(2)^4-6 t(2)^4-8 t(1) t(2)^3+9 t(2)^3+9 t(1) t(2)^2-8 t(2)^2-6 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{3}{q^{25/2}}+\frac{7}{q^{23/2}}-\frac{11}{q^{21/2}}+\frac{16}{q^{19/2}}-\frac{19}{q^{17/2}}+\frac{18}{q^{15/2}}-\frac{17}{q^{13/2}}+\frac{12}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{13} (-z)-a^{13} z^{-1} +3 a^{11} z^3+5 a^{11} z+a^{11} z^{-1} -2 a^9 z^5-3 a^9 z^3+a^9 z+2 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-6 a^7 z-2 a^7 z^{-1} -a^5 z^5-2 a^5 z^3-a^5 z (db)
Kauffman polynomial -z^6 a^{16}+3 z^4 a^{16}-3 z^2 a^{16}+a^{16}-3 z^7 a^{15}+8 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-4 z^8 a^{14}+7 z^6 a^{14}-3 z^2 a^{14}-3 z^9 a^{13}-z^7 a^{13}+13 z^5 a^{13}-9 z^3 a^{13}+a^{13} z^{-1} -z^{10} a^{12}-9 z^8 a^{12}+22 z^6 a^{12}-17 z^4 a^{12}+9 z^2 a^{12}-3 a^{12}-7 z^9 a^{11}+6 z^7 a^{11}+4 z^5 a^{11}-z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} -z^{10} a^{10}-11 z^8 a^{10}+22 z^6 a^{10}-14 z^4 a^{10}+3 z^2 a^{10}-4 z^9 a^9-2 z^7 a^9+11 z^5 a^9-11 z^3 a^9+7 z a^9-2 a^9 z^{-1} -6 z^8 a^8+5 z^6 a^8+4 z^4 a^8-7 z^2 a^8+3 a^8-6 z^7 a^7+11 z^5 a^7-11 z^3 a^7+7 z a^7-2 a^7 z^{-1} -3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5 (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          31-2
-8         5  5
-10        73  -4
-12       105   5
-14      98    -1
-16     109     1
-18    69      3
-20   510       -5
-22  26        4
-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}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a60

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L11a62