L11a193

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L11a192.gif

L11a192

L11a194.gif

L11a194

Contents

L11a193.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a193 at Knotilus!


Link Presentations

[edit Notes on L11a193's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X18,14,19,13 X20,16,21,15 X14,20,15,19 X12,22,13,21 X2738 X4,12,5,11 X6,18,1,17
Gauss code {1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 10, -8, 5, -7, 6, -4, 11, -5, 7, -6, 8, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a193 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-4 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-4 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial -14 q^{9/2}+9 q^{7/2}-6 q^{5/2}+2 q^{3/2}+q^{23/2}-4 q^{21/2}+8 q^{19/2}-12 q^{17/2}+16 q^{15/2}-17 q^{13/2}+16 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial z^5 a^{-9} +2 z^3 a^{-9} -z^7 a^{-7} -3 z^5 a^{-7} -z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -6 z^3 a^{-5} -6 z a^{-5} -3 a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +5 z a^{-3} +2 a^{-3} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -6 z^9 a^{-7} -4 z^9 a^{-9} -2 z^8 a^{-4} -4 z^8 a^{-6} -10 z^8 a^{-8} -8 z^8 a^{-10} -z^7 a^{-3} +2 z^7 a^{-5} +7 z^7 a^{-7} -6 z^7 a^{-9} -10 z^7 a^{-11} +7 z^6 a^{-4} +16 z^6 a^{-6} +22 z^6 a^{-8} +5 z^6 a^{-10} -8 z^6 a^{-12} +5 z^5 a^{-3} +12 z^5 a^{-5} +15 z^5 a^{-7} +24 z^5 a^{-9} +12 z^5 a^{-11} -4 z^5 a^{-13} -6 z^4 a^{-4} -8 z^4 a^{-6} -3 z^4 a^{-8} +8 z^4 a^{-10} +8 z^4 a^{-12} -z^4 a^{-14} -9 z^3 a^{-3} -23 z^3 a^{-5} -21 z^3 a^{-7} -14 z^3 a^{-9} -5 z^3 a^{-11} +2 z^3 a^{-13} -2 z^2 a^{-4} -6 z^2 a^{-6} -6 z^2 a^{-8} -5 z^2 a^{-10} -3 z^2 a^{-12} +7 z a^{-3} +13 z a^{-5} +8 z a^{-7} +3 z a^{-9} +z a^{-11} +3 a^{-4} +3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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 \
-2-10123456789χ
24           1-1
22          3 3
20         51 -4
18        73  4
16       95   -4
14      87    1
12     89     1
10    68      -2
8   49       5
6  25        -3
4 15         4
2 1          -1
01           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=4 i=6
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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See/edit the Link Page master template (intermediate).

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