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

Visit L11n416 at Knotilus!

Link Presentations

[edit Notes on L11n416's Link Presentations]

Planar diagram presentation X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code {1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, -9, 11, 8, -7, -10, 9, -11, 10}
A Braid Representative
A Morse Link Presentation L11n416 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v w^3-u^2 v w^2+u^2 v w-u^2 v-u^2 w^3+u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-2 u v w^3+2 u v w^2-2 u v w+2 u v+u w^3-u w^2+u w-u+v^2+v w^3-v w^2+v w-v}{u v w^{3/2}} (db)
Jones polynomial -2 q^{-9} +4 q^{-8} -6 q^{-7} +9 q^{-6} -8 q^{-5} +9 q^{-4} -6 q^{-3} +5 q^{-2} -2 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} -a^{10}+a^8 z^4+4 a^8 z^2+4 a^8 z^{-2} +6 a^8-a^6 z^6-4 a^6 z^4-7 a^6 z^2-5 a^6 z^{-2} -9 a^6-a^4 z^6-3 a^4 z^4-a^4 z^2+2 a^4 z^{-2} +2 a^4+a^2 z^4+3 a^2 z^2+2 a^2 (db)
Kauffman polynomial 3 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +a^{10} z^6+2 a^{10} z^4-5 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+4 a^9 z^7-15 a^9 z^5+30 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +4 a^8 z^8-16 a^8 z^6+34 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9+4 a^7 z^7-23 a^7 z^5+43 a^7 z^3-33 a^7 z+9 a^7 z^{-1} +6 a^6 z^8-22 a^6 z^6+36 a^6 z^4-36 a^6 z^2-5 a^6 z^{-2} +20 a^6+a^5 z^9+2 a^5 z^7-14 a^5 z^5+19 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-4 a^4 z^6-4 a^4 z^2-2 a^4 z^{-2} +6 a^4+2 a^3 z^7-6 a^3 z^5+3 a^3 z^3+a^3 z+a^2 z^6-4 a^2 z^4+5 a^2 z^2-2 a^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 \
1         11
-1        1 -1
-3       41 3
-5      43  -1
-7     52   3
-9    45    1
-11   54     1
-13  14      3
-15 35       -2
-17 2        2
-192         -2
Integral Khovanov Homology

(db, data source)

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

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