L11a181

From Knot Atlas
Jump to: navigation, search

L11a180.gif

L11a180

L11a182.gif

L11a182

Contents

L11a181.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a181 at Knotilus!


Link Presentations

[edit Notes on L11a181's Link Presentations]

Planar diagram presentation X8192 X20,9,21,10 X14,5,15,6 X16,8,17,7 X10,4,11,3 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X2,20,3,19
Gauss code {1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a181 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-7 u^2 v^3+8 u^2 v^2-3 u^2 v-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u-3 v^3+8 v^2-7 v+2}{u v^2} (db)
Jones polynomial -q^{11/2}+5 q^{9/2}-10 q^{7/2}+16 q^{5/2}-22 q^{3/2}+24 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+2 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+4 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -a^3 z+4 a z-6 z a^{-1} +2 z a^{-3} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -4 z^{10} a^{-2} -4 z^{10}-12 a z^9-20 z^9 a^{-1} -8 z^9 a^{-3} -17 a^2 z^8-4 z^8 a^{-2} -5 z^8 a^{-4} -16 z^8-15 a^3 z^7+12 a z^7+52 z^7 a^{-1} +24 z^7 a^{-3} -z^7 a^{-5} -9 a^4 z^6+29 a^2 z^6+39 z^6 a^{-2} +15 z^6 a^{-4} +62 z^6-4 a^5 z^5+21 a^3 z^5+18 a z^5-28 z^5 a^{-1} -19 z^5 a^{-3} +2 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-15 a^2 z^4-36 z^4 a^{-2} -12 z^4 a^{-4} -46 z^4+a^5 z^3-12 a^3 z^3-22 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} +6 z^2+2 a^3 z+9 a z+10 z a^{-1} +3 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} 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 \
-5-4-3-2-10123456χ
12           11
10          4 -4
8         61 5
6        104  -6
4       126   6
2      1210    -2
0     1312     1
-2    913      4
-4   712       -5
-6  310        7
-8 16         -5
-10 3          3
-121           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\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.

L11a180.gif

L11a180

L11a182.gif

L11a182