L11n438

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L11n437.gif

L11n437

L11n439.gif

L11n439

Contents

L11n438.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n438's Link Presentations]

Planar diagram presentation X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X19,13,20,22 X15,21,16,20 X21,17,22,16 X17,9,18,12
Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}
A Braid Representative
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A Morse Link Presentation L11n438 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(4)^3-t(1) t(2) t(4)^3+t(3) t(4)^3-t(4)^3-2 t(1) t(4)^2+3 t(1) t(2) t(4)^2-t(2) t(4)^2+t(1) t(3) t(4)^2-2 t(1) t(2) t(3) t(4)^2+2 t(2) t(3) t(4)^2-3 t(3) t(4)^2+2 t(4)^2+2 t(1) t(4)-3 t(1) t(2) t(4)+t(2) t(4)-t(1) t(3) t(4)+2 t(1) t(2) t(3) t(4)-2 t(2) t(3) t(4)+3 t(3) t(4)-2 t(4)+t(1) t(2)-t(1) t(2) t(3)+t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} (db)
Jones polynomial 11 q^{9/2}-13 q^{7/2}+6 q^{5/2}-4 q^{3/2}+q^{21/2}-2 q^{19/2}+6 q^{17/2}-10 q^{15/2}+12 q^{13/2}-15 q^{11/2} (db)
Signature 3 (db)
HOMFLY-PT polynomial -3 z^5 a^{-5} +4 z^3 a^{-3} -12 z^3 a^{-5} +6 z^3 a^{-7} +9 z a^{-3} -22 z a^{-5} +17 z a^{-7} -4 z a^{-9} +7 a^{-3} z^{-1} -20 a^{-5} z^{-1} +20 a^{-7} z^{-1} -8 a^{-9} z^{-1} + a^{-11} z^{-1} +2 a^{-3} z^{-3} -7 a^{-5} z^{-3} +9 a^{-7} z^{-3} -5 a^{-9} z^{-3} + a^{-11} z^{-3} (db)
Kauffman polynomial z^6 a^{-12} -4 z^4 a^{-12} +6 z^2 a^{-12} + a^{-12} z^{-2} -4 a^{-12} +2 z^7 a^{-11} -5 z^5 a^{-11} +4 z^3 a^{-11} - a^{-11} z^{-3} -2 z a^{-11} +2 a^{-11} z^{-1} +2 z^8 a^{-10} +2 z^6 a^{-10} -21 z^4 a^{-10} +34 z^2 a^{-10} +7 a^{-10} z^{-2} -24 a^{-10} +z^9 a^{-9} +8 z^7 a^{-9} -25 z^5 a^{-9} +22 z^3 a^{-9} -5 a^{-9} z^{-3} -16 z a^{-9} +12 a^{-9} z^{-1} +7 z^8 a^{-8} -46 z^4 a^{-8} +75 z^2 a^{-8} +18 a^{-8} z^{-2} -58 a^{-8} +z^9 a^{-7} +16 z^7 a^{-7} -46 z^5 a^{-7} +50 z^3 a^{-7} -9 a^{-7} z^{-3} -37 z a^{-7} +24 a^{-7} z^{-1} +5 z^8 a^{-6} +5 z^6 a^{-6} -41 z^4 a^{-6} +73 z^2 a^{-6} +19 a^{-6} z^{-2} -60 a^{-6} +10 z^7 a^{-5} -26 z^5 a^{-5} +42 z^3 a^{-5} -7 a^{-5} z^{-3} -39 z a^{-5} +23 a^{-5} z^{-1} +6 z^6 a^{-4} -12 z^4 a^{-4} +26 z^2 a^{-4} +7 a^{-4} z^{-2} -23 a^{-4} +10 z^3 a^{-3} -2 a^{-3} z^{-3} -16 z a^{-3} +9 a^{-3} 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 \
0123456789χ
22         1-1
20        1 1
18       51 -4
16      51  4
14     75   -2
12    85    3
10   711     4
8  64      2
6  7       7
446        -2
24         4
Integral Khovanov Homology

(db, data source)

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

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L11n437

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L11n439