L11a19

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L11a18.gif

L11a18

L11a20.gif

L11a20

Contents

L11a19.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a19 at Knotilus!


Link Presentations

[edit Notes on L11a19's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X20,14,21,13 X22,17,5,18 X18,21,19,22 X14,20,15,19 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 9, -4, 6, -7, 8, -5, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+4 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial 16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-16 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +9 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-6 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -8 z^9 a^{-3} -5 z^9 a^{-5} -10 z^8 a^{-2} -16 z^8 a^{-4} -9 z^8 a^{-6} -3 z^8-a z^7+4 z^7 a^{-1} +10 z^7 a^{-3} -4 z^7 a^{-5} -9 z^7 a^{-7} +41 z^6 a^{-2} +49 z^6 a^{-4} +13 z^6 a^{-6} -6 z^6 a^{-8} +11 z^6+4 a z^5+13 z^5 a^{-1} +25 z^5 a^{-3} +33 z^5 a^{-5} +14 z^5 a^{-7} -3 z^5 a^{-9} -42 z^4 a^{-2} -40 z^4 a^{-4} -4 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -13 z^4-6 a z^3-25 z^3 a^{-1} -40 z^3 a^{-3} -36 z^3 a^{-5} -13 z^3 a^{-7} +2 z^3 a^{-9} +15 z^2 a^{-2} +11 z^2 a^{-4} -4 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +5 z^2+4 a z+13 z a^{-1} +20 z a^{-3} +16 z a^{-5} +5 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         51 -4
12        72  5
10       95   -4
8      97    2
6     89     1
4    89      -1
2   510       5
0  26        -4
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology

(db, data source)

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

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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