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

Visit L11n312 at Knotilus!

Link Presentations

[edit Notes on L11n312's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -4, 9, -3, 8}, {-11, 2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}
A Braid Representative
A Morse Link Presentation L11n312 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w^3-2 u v^2 w^2+2 u v^2 w-u v^2+u v w^4-2 u v w^3+3 u v w^2-3 u v w+u v+u w-v^2 w^3-v w^4+3 v w^3-3 v w^2+2 v w-v+w^4-2 w^3+2 w^2-w}{\sqrt{u} v w^2} (db)
Jones polynomial 1-3 q^{-1} +7 q^{-2} -8 q^{-3} +12 q^{-4} -11 q^{-5} +11 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} -a^{10}+z^4 a^8+4 z^2 a^8+4 a^8 z^{-2} +7 a^8-z^6 a^6-4 z^4 a^6-9 z^2 a^6-5 a^6 z^{-2} -12 a^6-z^6 a^4-2 z^4 a^4+2 z^2 a^4+2 a^4 z^{-2} +5 a^4+z^4 a^2+2 z^2 a^2+a^2 (db)
Kauffman polynomial 3 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} +z^6 a^{10}+4 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+5 z^7 a^9-15 z^5 a^9+28 z^3 a^9-19 z a^9+5 a^9 z^{-1} +6 z^8 a^8-21 z^6 a^8+39 z^4 a^8-32 z^2 a^8-4 a^8 z^{-2} +15 a^8+2 z^9 a^7+4 z^7 a^7-27 z^5 a^7+46 z^3 a^7-33 z a^7+9 a^7 z^{-1} +10 z^8 a^6-33 z^6 a^6+44 z^4 a^6-37 z^2 a^6-5 a^6 z^{-2} +20 a^6+2 z^9 a^5+2 z^7 a^5-20 z^5 a^5+25 z^3 a^5-18 z a^5+5 a^5 z^{-1} +4 z^8 a^4-10 z^6 a^4+6 z^4 a^4-8 z^2 a^4-2 a^4 z^{-2} +8 a^4+3 z^7 a^3-8 z^5 a^3+4 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^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        2 -2
-3       51 4
-5      54  -1
-7     73   4
-9    56    1
-11   66     0
-13  25      3
-15 36       -3
-17 3        3
-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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