From Knot Atlas
Jump to: navigation, search






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

Visit L11a148 at Knotilus!

Link Presentations

[edit Notes on L11a148's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X20,11,21,12 X12,6,13,5 X4,20,5,19 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22
Gauss code {1, -2, 3, -6, 5, -11}, {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}
A Braid Representative
A Morse Link Presentation L11a148 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-5 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+6 u v^3-9 u v^2+6 u v-2 u-2 v^3+5 v^2-5 v+2}{u v^2} (db)
Jones polynomial 4 q^{9/2}-\frac{3}{q^{9/2}}-7 q^{7/2}+\frac{6}{q^{7/2}}+11 q^{5/2}-\frac{11}{q^{5/2}}-15 q^{3/2}+\frac{14}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-z^5 a^{-3} -3 a^3 z^3-2 z^3 a^{-3} -2 a^3 z+z a^{-3} + a^{-3} z^{-1} +a z^7+z^7 a^{-1} +4 a z^5+3 z^5 a^{-1} +6 a z^3+5 a z-5 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-3 z^5 a^{-5} -3 a^5 z^3+2 z^3 a^{-5} +a^5 z+4 z^8 a^{-4} +5 a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+15 z^4 a^{-4} +a^4 z^2-2 z^2 a^{-4} - a^{-4} +5 z^9 a^{-3} +7 a^3 z^7-17 z^7 a^{-3} -9 a^3 z^5+15 z^5 a^{-3} +6 a^3 z^3-3 z^3 a^{-3} -a^3 z-z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +7 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-26 z^6 a^{-2} +2 a^2 z^4+25 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} -3 a^{-2} +5 a z^9+10 z^9 a^{-1} -3 a z^7-28 z^7 a^{-1} -12 a z^5+18 z^5 a^{-1} +15 a z^3+z^3 a^{-1} -8 a z-7 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}+6 z^8-25 z^6+17 z^4-3 (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 \
12           11
10          3 -3
8         41 3
6        73  -4
4       84   4
2      87    -1
0     98     1
-2    69      3
-4   58       -3
-6  27        5
-8 14         -3
-10 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.