L11a284

From Knot Atlas
Jump to: navigation, search

L11a283.gif

L11a283

L11a285.gif

L11a285

Contents

L11a284.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a284 at Knotilus!


Link Presentations

[edit Notes on L11a284's Link Presentations]

Planar diagram presentation X10,1,11,2 X22,11,9,12 X8,9,1,10 X2,22,3,21 X14,6,15,5 X12,4,13,3 X4,14,5,13 X16,20,17,19 X18,8,19,7 X6,18,7,17 X20,16,21,15
Gauss code {1, -4, 6, -7, 5, -10, 9, -3}, {3, -1, 2, -6, 7, -5, 11, -8, 10, -9, 8, -11, 4, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a284 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v+1) (u v-u-v+2) (2 u v-u-v+1)}{u^{3/2} v^{3/2}} (db)
Jones polynomial 16 q^{9/2}-16 q^{7/2}+13 q^{5/2}-11 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-6 q^{15/2}+10 q^{13/2}-14 q^{11/2}+6 \sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} +z^7 a^{-5} -z^5 a^{-1} +4 z^5 a^{-3} +4 z^5 a^{-5} -z^5 a^{-7} -3 z^3 a^{-1} +6 z^3 a^{-3} +5 z^3 a^{-5} -3 z^3 a^{-7} -2 z a^{-1} +5 z a^{-3} +z a^{-5} -2 z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +3 z^2 a^{-10} +4 z^7 a^{-9} -5 z^5 a^{-9} +z a^{-9} +4 z^8 a^{-8} -3 z^6 a^{-8} -z^4 a^{-8} +3 z^9 a^{-7} -z^7 a^{-7} +z^5 a^{-7} -4 z^3 a^{-7} +3 z a^{-7} +z^{10} a^{-6} +6 z^8 a^{-6} -13 z^6 a^{-6} +11 z^4 a^{-6} -3 z^2 a^{-6} +6 z^9 a^{-5} -10 z^7 a^{-5} +7 z^5 a^{-5} -3 z^3 a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +6 z^8 a^{-4} -17 z^6 a^{-4} +12 z^4 a^{-4} -z^2 a^{-4} - a^{-4} +3 z^9 a^{-3} -2 z^7 a^{-3} -9 z^5 a^{-3} +11 z^3 a^{-3} -6 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -9 z^6 a^{-2} +3 z^4 a^{-2} +z^2 a^{-2} +3 z^7 a^{-1} -9 z^5 a^{-1} +8 z^3 a^{-1} -3 z a^{-1} +z^6-3 z^4+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 \
-3-2-1012345678χ
20           11
18          2 -2
16         41 3
14        62  -4
12       84   4
10      97    -2
8     77     0
6    69      3
4   57       -2
2  27        5
0 14         -3
-2 2          2
-41           -1
Integral Khovanov Homology

(db, data source)

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

L11a283.gif

L11a283

L11a285.gif

L11a285