L11a429

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L11a428.gif

L11a428

L11a430.gif

L11a430

Contents

L11a429.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a429's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X22,20,11,19 X8,22,9,21 X20,8,21,7 X2,9,3,10 X4,18,1,17
Gauss code {1, -10, 2, -11}, {5, -1, 9, -8, 10, -6}, {3, -2, 4, -5, 6, -4, 11, -3, 7, -9, 8, -7}
A Braid Representative
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A Morse Link Presentation L11a429 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-2 t(2)^2-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-5 t(1) t(2)+9 t(1) t(3) t(2)-9 t(3) t(2)+4 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-4 t(1) t(3)+5 t(3)-2}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial q^9-4 q^8+9 q^7-14 q^6+20 q^5-22 q^4+23 q^3-19 q^2- q^{-2} +15 q+4 q^{-1} -8 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} -2 z^4 a^{-6} -z^4+4 z^2 a^{-2} -4 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} -z^2+4 a^{-2} -6 a^{-4} +2 a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} (db)
Kauffman polynomial z^6 a^{-10} -2 z^4 a^{-10} +z^2 a^{-10} +4 z^7 a^{-9} -9 z^5 a^{-9} +5 z^3 a^{-9} +7 z^8 a^{-8} -16 z^6 a^{-8} +11 z^4 a^{-8} -4 z^2 a^{-8} + a^{-8} +6 z^9 a^{-7} -7 z^7 a^{-7} -4 z^5 a^{-7} +3 z^3 a^{-7} +2 z^{10} a^{-6} +12 z^8 a^{-6} -34 z^6 a^{-6} +23 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +12 z^9 a^{-5} -18 z^7 a^{-5} +6 z^5 a^{-5} -5 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +13 z^8 a^{-4} -28 z^6 a^{-4} +12 z^4 a^{-4} +8 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +6 z^9 a^{-3} -9 z^5 a^{-3} +z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +8 z^8 a^{-2} -7 z^6 a^{-2} -4 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2} +7 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-6 z^4+3 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 \
-3-2-1012345678χ
19           11
17          3 -3
15         61 5
13        94  -5
11       115   6
9      119    -2
7     1211     1
5    812      4
3   711       -4
1  310        7
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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L11a428

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L11a430