L11n72

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L11n71.gif

L11n71

L11n73.gif

L11n73

Contents

L11n72.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n72 at Knotilus!


Link Presentations

[edit Notes on L11n72's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X15,8,16,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4}
A Braid Representative
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A Morse Link Presentation L11n72 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-5 u v^2+6 u v-2 u-2 v^3+6 v^2-5 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+5 q^{5/2}-8 q^{3/2}+10 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^{-1} +a^3 z^3+z^3 a^{-3} -a^3 z-a^3 z^{-1} +z a^{-3} -a z^5-z^5 a^{-1} -a z^3-2 z^3 a^{-1} -2 z a^{-1} (db)
Kauffman polynomial 3 a^5 z^3-3 a^5 z+a^5 z^{-1} +a^4 z^6+z^6 a^{-4} +2 a^4 z^4-3 z^4 a^{-4} +a^4 z^2+2 z^2 a^{-4} -a^4+2 a^3 z^7+3 z^7 a^{-3} +a^3 z^5-10 z^5 a^{-3} +9 z^3 a^{-3} -a^3 z-z a^{-3} +a^3 z^{-1} +2 a^2 z^8+3 z^8 a^{-2} +a^2 z^6-7 z^6 a^{-2} -2 a^2 z^4+2 z^4 a^{-2} +z^2 a^{-2} +a z^9+z^9 a^{-1} +3 a z^7+4 z^7 a^{-1} -4 a z^5-15 z^5 a^{-1} -3 a z^3+9 z^3 a^{-1} +2 a z-z a^{-1} +5 z^8-8 z^6+z^4-2 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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       31 -2
4      52  3
2     53   -2
0    55    0
-2   56     1
-4  34      -1
-6 15       4
-813        -2
-102         2
Integral Khovanov Homology

(db, data source)

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