L11n307

From Knot Atlas
Jump to: navigation, search

L11n306.gif

L11n306

L11n308.gif

L11n308

Contents

L11n307.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n307 at Knotilus!


Link Presentations

[edit Notes on L11n307's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 5, -4, 6}, {-11, 2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11n307 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-3 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+3 u v w^2-2 u v w+u w^3-u w^2+v^2 w^2-v^2 w+2 v w^3-3 v w^2+v w+w^4-3 w^3+3 w^2}{\sqrt{u} v w^2} (db)
Jones polynomial -q^{10}+3 q^9-6 q^8+8 q^7-9 q^6+11 q^5-8 q^4+8 q^3-4 q^2+2 q (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-4} -3 z^4 a^{-6} +2 z^2 a^{-2} +2 z^2 a^{-4} -9 z^2 a^{-6} +4 z^2 a^{-8} + a^{-2} +5 a^{-4} -12 a^{-6} +7 a^{-8} - a^{-10} +2 a^{-4} z^{-2} -5 a^{-6} z^{-2} +4 a^{-8} z^{-2} - a^{-10} z^{-2} (db)
Kauffman polynomial z^7 a^{-11} -4 z^5 a^{-11} +6 z^3 a^{-11} -4 z a^{-11} + a^{-11} z^{-1} +3 z^8 a^{-10} -12 z^6 a^{-10} +14 z^4 a^{-10} -6 z^2 a^{-10} - a^{-10} z^{-2} +3 a^{-10} +2 z^9 a^{-9} -z^7 a^{-9} -21 z^5 a^{-9} +36 z^3 a^{-9} -19 z a^{-9} +5 a^{-9} z^{-1} +10 z^8 a^{-8} -39 z^6 a^{-8} +48 z^4 a^{-8} -32 z^2 a^{-8} -4 a^{-8} z^{-2} +15 a^{-8} +2 z^9 a^{-7} +5 z^7 a^{-7} -38 z^5 a^{-7} +52 z^3 a^{-7} -33 z a^{-7} +9 a^{-7} z^{-1} +7 z^8 a^{-6} -24 z^6 a^{-6} +35 z^4 a^{-6} -37 z^2 a^{-6} -5 a^{-6} z^{-2} +20 a^{-6} +7 z^7 a^{-5} -20 z^5 a^{-5} +25 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +3 z^6 a^{-4} +z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +z^5 a^{-3} +3 z^3 a^{-3} +3 z^2 a^{-2} - a^{-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 \
0123456789χ
21         1-1
19        2 2
17       41 -3
15      42  2
13     65   -1
11    53    2
9   36     3
7  55      0
5  4       4
324        -2
12         2
Integral Khovanov Homology

(db, data source)

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

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.

L11n306.gif

L11n306

L11n308.gif

L11n308