From Knot Atlas
Jump to: navigation, search






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

Visit L11a520 at Knotilus!

Link Presentations

[edit Notes on L11a520's Link Presentations]

Planar diagram presentation X8192 X18,8,19,7 X16,6,17,5 X10,16,11,15 X14,3,15,4 X4,11,5,12 X2,20,3,19 X20,9,21,10 X12,13,7,14 X22,18,13,17 X6,21,1,22
Gauss code {1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}
A Braid Representative
A Morse Link Presentation L11a520 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u v-u w^2+2 u w-u-v w^2+2 v w-v+w^2-w+1\right)}{u v w^{3/2}} (db)
Jones polynomial -q^5+5 q^4-13 q^3+23 q^2-31 q+37-36 q^{-1} +33 q^{-2} -23 q^{-3} +15 q^{-4} -6 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+a^4 z^{-2} -2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2+z^8+4 z^6+7 z^4+4 z^2+ z^{-2} +1 (db)
Kauffman polynomial 7 a^2 z^{10}+7 z^{10}+17 a^3 z^9+36 a z^9+19 z^9 a^{-1} +15 a^4 z^8+22 a^2 z^8+21 z^8 a^{-2} +28 z^8+6 a^5 z^7-27 a^3 z^7-67 a z^7-21 z^7 a^{-1} +13 z^7 a^{-3} +a^6 z^6-27 a^4 z^6-72 a^2 z^6-32 z^6 a^{-2} +5 z^6 a^{-4} -81 z^6-6 a^5 z^5+6 a^3 z^5+26 a z^5-13 z^5 a^{-3} +z^5 a^{-5} +11 a^4 z^4+41 a^2 z^4+21 z^4 a^{-2} -2 z^4 a^{-4} +53 z^4+a^3 z^3-a z^3+3 z^3 a^{-1} +5 z^3 a^{-3} +2 a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -10 z^2-a^3 z-a z+a^4+a^2+1+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 \
11           1-1
9          4 4
7         91 -8
5        144  10
3       179   -8
1      2014    6
-1     1819     1
-3    1518      -3
-5   1020       10
-7  513        -8
-9 110         9
-11 5          -5
-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}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{20}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=-1 {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{18} {\mathbb Z}^{18}
r=0 {\mathbb Z}^{19}\oplus{\mathbb Z}_2^{18} {\mathbb Z}^{20}
r=1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{17} {\mathbb Z}^{17}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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.