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

Visit L11a454 at Knotilus!

Link Presentations

[edit Notes on L11a454's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,18,13,17 X16,9,17,10 X10,15,11,16 X12,20,5,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 6, -7, 3, -8}, {11, -2, 7, -6, 5, -4, 8, -3, 9, -5}
A Braid Representative
A Morse Link Presentation L11a454 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-t(3)^2 t(2)^3+t(1) t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+7 t(3)^2 t(2)^2-3 t(1) t(2)^2+6 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+3 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+2 t(1) t(2)-7 t(1) t(3) t(2)+4 t(3) t(2)-t(2)+t(1) t(3)^3-t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -q^5-3 q^{-5} +4 q^4+8 q^{-4} -10 q^3-14 q^{-3} +17 q^2+21 q^{-2} -21 q-23 q^{-1} +25 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-3 a^4 z^2+a^4 z^{-2} -z^2 a^{-4} -a^4+3 a^2 z^4+2 z^4 a^{-2} +a^2 z^2-2 a^2 z^{-2} -3 a^2- a^{-2} -z^6+2 z^2+ z^{-2} +4 (db)
Kauffman polynomial 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+13 a z^9+8 z^9 a^{-1} +5 a^4 z^8+13 a^2 z^8+12 z^8 a^{-2} +20 z^8+3 a^5 z^7-3 a^3 z^7-16 a z^7-z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-9 a^4 z^6-39 a^2 z^6-20 z^6 a^{-2} +4 z^6 a^{-4} -53 z^6-7 a^5 z^5-10 a^3 z^5-8 a z^5-19 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+5 a^4 z^4+41 a^2 z^4+16 z^4 a^{-2} -4 z^4 a^{-4} +53 z^4+5 a^5 z^3+14 a^3 z^3+19 a z^3+19 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2-3 a^4 z^2-27 a^2 z^2-10 z^2 a^{-2} +z^2 a^{-4} -32 z^2-a^5 z-8 a^3 z-12 a z-7 z a^{-1} -2 z a^{-3} -a^6+3 a^4+11 a^2+3 a^{-2} +11+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         71 -6
5        103  7
3       117   -4
1      1410    4
-1     1214     2
-3    911      -2
-5   512       7
-7  39        -6
-9 16         5
-11 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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