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

Visit L11n275 at Knotilus!

Link Presentations

[edit Notes on L11n275's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X19,10,20,5
Gauss code {1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, 4, -5, -10, 9, -7, 6, -11, 8, -9, 7}
A Braid Representative
A Morse Link Presentation L11n275 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w^4-3 u v w^3+3 u v w^2-u v w+2 u w^3-2 u w^2+u w-v^2 w^3+2 v^2 w^2-2 v^2 w+v w^3-3 v w^2+3 v w-v}{\sqrt{u} v w^2} (db)
Jones polynomial -2 q^3+4 q^2-6 q+9-8 q^{-1} +9 q^{-2} -6 q^{-3} +5 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-5 a^2 z^4+3 z^4+3 a^4 z^2-12 a^2 z^2-2 z^2 a^{-2} +10 z^2+4 a^4-13 a^2-3 a^{-2} +12+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+2 a^5 z^7+3 a^3 z^7+4 a z^7+3 z^7 a^{-1} +a^6 z^6-11 a^4 z^6-22 a^2 z^6+z^6 a^{-2} -9 z^6-6 a^5 z^5-20 a^3 z^5-23 a z^5-9 z^5 a^{-1} -4 a^6 z^4+15 a^4 z^4+36 a^2 z^4+2 z^4 a^{-2} +19 z^4+3 a^5 z^3+28 a^3 z^3+43 a z^3+21 z^3 a^{-1} +3 z^3 a^{-3} +4 a^6 z^2-15 a^4 z^2-36 a^2 z^2-4 z^2 a^{-2} -21 z^2-18 a^3 z-33 a z-19 z a^{-1} -4 z a^{-3} -a^6+8 a^4+20 a^2+3 a^{-2} +15+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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 \
7         2-2
5        2 2
3       42 -2
1      52  3
-1     56   1
-3    43    1
-5   36     3
-7  23      -1
-9  3       3
-1112        -1
-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} {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}_2^{2} {\mathbb Z}^{2}

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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