L11n128

From Knot Atlas
Jump to: navigation, search

L11n127.gif

L11n127

L11n129.gif

L11n129

Contents

L11n128.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n128 at Knotilus!


Link Presentations

[edit Notes on L11n128's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X16,12,17,11 X6,18,1,17 X19,14,20,15 X13,20,14,21 X21,19,22,18
Gauss code {1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 11, -9, 10, -11, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n128 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-3 u^2 v^3+2 u^2 v^2-u v^4+2 u v^3-u v^2+2 u v-u+2 v^2-3 v+1}{u v^2} (db)
Jones polynomial 4 q^{9/2}-6 q^{7/2}+6 q^{5/2}-\frac{1}{q^{5/2}}-6 q^{3/2}+\frac{2}{q^{3/2}}+q^{13/2}-3 q^{11/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-8 z^3 a^{-1} +8 z^3 a^{-3} -4 z^3 a^{-5} +3 a z-7 z a^{-1} +7 z a^{-3} -3 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -3 z^8 a^{-4} -2 z^8-a z^7-2 z^7 a^{-3} -3 z^7 a^{-5} +20 z^6 a^{-2} +10 z^6 a^{-4} -z^6 a^{-6} +9 z^6+5 a z^5+15 z^5 a^{-1} +20 z^5 a^{-3} +10 z^5 a^{-5} -20 z^4 a^{-2} -9 z^4 a^{-4} -11 z^4-8 a z^3-25 z^3 a^{-1} -27 z^3 a^{-3} -13 z^3 a^{-5} -3 z^3 a^{-7} +6 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-8} +4 z^2+5 a z+13 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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 \
-4-3-2-1012345χ
14         1-1
12        2 2
10       21 -1
8      42  2
6     33   0
4    33    0
2   34     1
0  12      -1
-2 13       2
-4 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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.

L11n127.gif

L11n127

L11n129.gif

L11n129