L11a437

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L11a436.gif

L11a436

L11a438.gif

L11a438

Contents

L11a437.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a437 at Knotilus!


Link Presentations

[edit Notes on L11a437's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,11,19,12 X14,8,15,7 X8,14,9,13 X22,15,13,16 X20,17,21,18 X16,21,17,22 X12,19,5,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, 6, -8, 7, -3, 9, -7, 8, -6}
A Braid Representative
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A Morse Link Presentation L11a437 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) (w-1) \left(3 u v-2 u+2 v^2-3 v\right)}{\sqrt{u} v^{3/2} \sqrt{w}} (db)
Jones polynomial - q^{-10} +3 q^{-9} -6 q^{-8} +9 q^{-7} -10 q^{-6} +13 q^{-5} -12 q^{-4} +11 q^{-3} -7 q^{-2} +q+5 q^{-1} -2 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^{10}+3 a^8 z^2+2 a^8-2 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -2 a^4 z^4-2 a^4 z^2-2 a^4 z^{-2} -3 a^4-a^2 z^4+a^2 z^{-2} +a^2+z^2+1 (db)
Kauffman polynomial z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-12 z^6 a^{10}+14 z^4 a^{10}-7 z^2 a^{10}+2 a^{10}+3 z^9 a^9-8 z^7 a^9-2 z^5 a^9+11 z^3 a^9-4 z a^9+z^{10} a^8+5 z^8 a^8-30 z^6 a^8+38 z^4 a^8-18 z^2 a^8+4 a^8+6 z^9 a^7-17 z^7 a^7+10 z^5 a^7-z^3 a^7+z^{10} a^6+5 z^8 a^6-21 z^6 a^6+19 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -3 a^6+3 z^9 a^5-5 z^7 a^5+6 z^5 a^5-10 z^3 a^5+8 z a^5-2 a^5 z^{-1} +3 z^8 a^4-9 z^4 a^4+13 z^2 a^4+2 a^4 z^{-2} -8 a^4+3 z^7 a^3-5 z^3 a^3+6 z a^3-2 a^3 z^{-1} +3 z^6 a^2-3 z^4 a^2+3 z^2 a^2+a^2 z^{-2} -3 a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+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 \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          1 -1
-1         41 3
-3        53  -2
-5       62   4
-7      65    -1
-9     76     1
-11    58      3
-13   45       -1
-15  25        3
-17 14         -3
-19 2          2
-211           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a436.gif

L11a436

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L11a438