L11a474

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L11a473.gif

L11a473

L11a475.gif

L11a475

Contents

L11a474.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a474 at Knotilus!


Link Presentations

[edit Notes on L11a474's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X20,14,21,13 X8,19,9,20 X18,9,19,10 X22,16,17,15 X16,18,5,17 X14,22,15,21 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {8, -6, 5, -4, 9, -7}, {10, -1, 3, -5, 6, -2, 11, -3, 4, -9, 7, -8}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a474 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(2)^2 t(3)^2+t(2) t(3)^2-t(3)^2+t(2)^2 t(3)-t(2) t(3)+t(3)+t(2)-1\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^9+3 q^8-6 q^7+9 q^6-11 q^5+13 q^4-11 q^3+11 q^2-7 q+5-2 q^{-1} + q^{-2} (db)
Signature 4 (db)
HOMFLY-PT polynomial -z^6 a^{-6} -4 z^4 a^{-6} -5 z^2 a^{-6} -3 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +14 z^4 a^{-4} +17 z^2 a^{-4} + a^{-4} z^{-2} +9 a^{-4} -2 z^6 a^{-2} -10 z^4 a^{-2} -16 z^2 a^{-2} -2 a^{-2} z^{-2} -10 a^{-2} +z^4+4 z^2+ z^{-2} +4 (db)
Kauffman polynomial z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +2 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8 a^{-6} +z^8-10 z^7 a^{-1} -28 z^7 a^{-3} -8 z^7 a^{-5} +10 z^7 a^{-7} -29 z^6 a^{-2} -54 z^6 a^{-4} -22 z^6 a^{-6} +9 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +27 z^5 a^{-3} -13 z^5 a^{-5} -19 z^5 a^{-7} +6 z^5 a^{-9} +61 z^4 a^{-2} +76 z^4 a^{-4} +12 z^4 a^{-6} -13 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-5 z^3 a^{-1} +3 z^3 a^{-3} +19 z^3 a^{-5} +6 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -47 z^2 a^{-2} -44 z^2 a^{-4} -5 z^2 a^{-6} +5 z^2 a^{-8} -13 z^2-4 z a^{-1} -9 z a^{-3} -6 z a^{-5} +z a^{-9} +15 a^{-2} +13 a^{-4} +2 a^{-6} - a^{-8} +6+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         41 -3
13        52  3
11       64   -2
9      75    2
7     68     2
5    55      0
3   48       4
1  13        -2
-1 14         3
-3 1          -1
-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}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a473.gif

L11a473

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L11a475