Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L11a19

From Knot Atlas
Jump to: navigation, search

L11a18.gif

L11a18

L11a20.gif

L11a20

Contents

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

Visit L11a19's page at Knotilus.

Visit L11a19's page at the original Knot Atlas.


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). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=3 is the signature of L11a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ 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).

Back to the top.

L11a18.gif

L11a18

L11a20.gif

L11a20