From Knot Atlas
Jump to: navigation, search






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

Visit L11a402 at Knotilus!

Link Presentations

[edit Notes on L11a402's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,22,5,21 X18,11,19,12 X16,9,17,10 X22,17,11,18 X8,19,9,20 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 7, -5}, {6, -2, 11, -4, 3, -7, 8, -6, 9, -3, 5, -8}
A Braid Representative
A Morse Link Presentation L11a402 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-7 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+7 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2} (db)
Jones polynomial -q^5+4 q^4-9 q^3+16 q^2-20 q+25-23 q^{-1} +21 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-8 a^2 z^2-3 z^2 a^{-2} +8 z^2+a^4-2 a^2+1+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial a^6 z^6-2 a^6 z^4+a^6 z^2+4 a^5 z^7-9 a^5 z^5+z^5 a^{-5} +6 a^5 z^3-z^3 a^{-5} -2 a^5 z+7 a^4 z^8-15 a^4 z^6+4 z^6 a^{-4} +10 a^4 z^4-5 z^4 a^{-4} -6 a^4 z^2+z^2 a^{-4} +2 a^4+6 a^3 z^9-3 a^3 z^7+8 z^7 a^{-3} -19 a^3 z^5-11 z^5 a^{-3} +21 a^3 z^3+4 z^3 a^{-3} -7 a^3 z-z a^{-3} +2 a^2 z^{10}+17 a^2 z^8+11 z^8 a^{-2} -56 a^2 z^6-20 z^6 a^{-2} +59 a^2 z^4+18 z^4 a^{-2} -28 a^2 z^2-9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +4 a^2+14 a z^9+8 z^9 a^{-1} -20 a z^7-5 z^7 a^{-1} -9 a z^5-11 z^5 a^{-1} +25 a z^3+15 z^3 a^{-1} -7 a z-3 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+21 z^8-64 z^6+70 z^4-31 z^2+2 z^{-2} +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 \
11           1-1
9          3 3
7         61 -5
5        103  7
3       117   -4
1      149    5
-1     1113     2
-3    1012      -2
-5   612       6
-7  39        -6
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.