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

Visit L11a459 at Knotilus!

Link Presentations

[edit Notes on L11a459's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X22,20,13,19 X18,8,19,7 X10,14,11,13 X8,21,9,22 X20,9,21,10 X2,11,3,12 X4,18,1,17
Gauss code {1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}
A Braid Representative
A Morse Link Presentation L11a459 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+5 t(1) t(3)^2 t(2)^2-5 t(3)^2 t(2)^2+t(1) t(2)^2-4 t(1) t(3) t(2)^2+5 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-t(3)^3 t(2)-5 t(1) t(3)^2 t(2)+4 t(3)^2 t(2)-2 t(1) t(2)+5 t(1) t(3) t(2)-5 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -q^5-4 q^{-5} +4 q^4+9 q^{-4} -8 q^3-13 q^{-3} +13 q^2+19 q^{-2} -18 q-20 q^{-1} +22 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^6-z^6 a^{-2} -7 a^2 z^4-3 z^4 a^{-2} -7 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -3 a^2+z^8+5 z^6+9 z^4+6 z^2+ z^{-2} +2 (db)
Kauffman polynomial 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+12 a z^9+6 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+8 z^8 a^{-2} +12 z^8+4 a^5 z^7-9 a^3 z^7-24 a z^7-4 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-17 a^4 z^6-39 a^2 z^6-12 z^6 a^{-2} +4 z^6 a^{-4} -37 z^6-9 a^5 z^5-2 a^3 z^5+14 a z^5-5 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+40 a^2 z^4+7 z^4 a^{-2} -6 z^4 a^{-4} +39 z^4+4 a^5 z^3+4 a^3 z^3+a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-7 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} -14 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+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          3 3
7         51 -4
5        83  5
3       105   -5
1      128    4
-1     1113     2
-3    89      -1
-5   511       6
-7  48        -4
-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}^{4}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

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