L11a442

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L11a441.gif

L11a441

L11a443.gif

L11a443

Contents

L11a442.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a442 at Knotilus!


Link Presentations

[edit Notes on L11a442's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X22,16,13,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X20,12,21,11 X10,20,11,19 X12,22,5,21 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 5, -4, 6, -8, 7, -9}, {11, -2, 3, -5, 4, -6, 8, -7, 9, -3}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a442 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^3 w^3-u v^3 w^2-u v^2 w^3+2 u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^3 w^3+v^3 w^2+v^2 w^3-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial -q^9+3 q^8-4 q^7+6 q^6-7 q^5+8 q^4-7 q^3+6 q^2-4 q+4- q^{-1} + q^{-2} (db)
Signature 4 (db)
HOMFLY-PT polynomial -z^6 a^{-6} -4 z^4 a^{-6} -3 z^2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +12 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +5 a^{-4} -2 z^6 a^{-2} -11 z^4 a^{-2} -18 z^2 a^{-2} -2 a^{-2} z^{-2} -11 a^{-2} +z^4+5 z^2+ z^{-2} +6 (db)
Kauffman polynomial z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +4 z^9 a^{-3} +3 z^9 a^{-5} -3 z^8 a^{-2} +4 z^8 a^{-6} +z^8-4 z^7 a^{-1} -18 z^7 a^{-3} -10 z^7 a^{-5} +4 z^7 a^{-7} -5 z^6 a^{-2} -13 z^6 a^{-4} -11 z^6 a^{-6} +4 z^6 a^{-8} -7 z^6+18 z^5 a^{-3} +8 z^5 a^{-5} -6 z^5 a^{-7} +4 z^5 a^{-9} +22 z^4 a^{-2} +19 z^4 a^{-4} +8 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4 a^{-10} +17 z^4+12 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} -3 z^3 a^{-9} +z^3 a^{-11} -25 z^2 a^{-2} -10 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} -18 z^2-11 z a^{-1} -11 z a^{-3} +13 a^{-2} +5 a^{-4} + a^{-8} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-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 \
-4-3-2-101234567χ
19           1-1
17          2 2
15         21 -1
13        42  2
11       54   -1
9      32    1
7     45     1
5    23      -1
3   35       2
1  11        0
-1  3         3
-311          0
-51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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L11a441.gif

L11a441

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L11a443