L11a221

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L11a220.gif

L11a220

L11a222.gif

L11a222

Contents

L11a221.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a221 at Knotilus!


Link Presentations

[edit Notes on L11a221's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X16,6,17,5 X20,13,21,14 X18,15,19,16 X14,19,15,20 X22,10,7,9 X4,18,5,17 X10,22,11,21 X2738 X6,11,1,12
Gauss code {1, -10, 2, -8, 3, -11}, {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 6, -4, 9, -7}
A Braid Representative
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A Morse Link Presentation L11a221 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u v-2 u-2 v+2) (2 u v-2 u-2 v+1)}{u v} (db)
Jones polynomial -\frac{10}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+3 q^{5/2}-\frac{16}{q^{5/2}}-6 q^{3/2}+\frac{15}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+10 \sqrt{q}-\frac{14}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+2 a^5 z^3+a^5 z-a^3 z^5+a^3 z^{-1} -z a^{-3} -a z^5+2 z^3 a^{-1} -a z-a z^{-1} +z a^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-4 a^6 z^8-6 a^4 z^8-6 a^2 z^8-4 z^8-3 a^7 z^7+3 a^5 z^7+11 a^3 z^7+a z^7-4 z^7 a^{-1} -a^8 z^6+10 a^6 z^6+20 a^4 z^6+15 a^2 z^6-3 z^6 a^{-2} +3 z^6+9 a^7 z^5+6 a^5 z^5-8 a^3 z^5+a z^5+5 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-6 a^6 z^4-20 a^4 z^4-15 a^2 z^4+6 z^4 a^{-2} +2 z^4-7 a^7 z^3-6 a^5 z^3+3 a^3 z^3+2 z^3 a^{-3} -2 a^8 z^2+2 a^6 z^2+7 a^4 z^2+5 a^2 z^2-3 z^2 a^{-2} -z^2+2 a^7 z+a^5 z-3 a^3 z-a z-z a^{-3} -a^2+a^3 z^{-1} +a 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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         41 3
2        62  -4
0       84   4
-2      87    -1
-4     87     1
-6    69      3
-8   47       -3
-10  26        4
-12 14         -3
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
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}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a220.gif

L11a220

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L11a222