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

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Link Presentations

[edit Notes on L10a163's Link Presentations]

Planar diagram presentation X8192 X20,10,13,9 X6,20,1,19 X18,7,19,8 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,17,7,18 X2,13,3,14 X14,3,15,4
Gauss code {1, -9, 10, -5, 6, -3}, {4, -1, 2, -7, 5, -8}, {9, -10, 7, -6, 8, -4, 3, -2}
A Braid Representative
A Morse Link Presentation L10a163 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u w+v w-w+1\right)}{u v w^{3/2}} (db)
Jones polynomial  q^{-7} -4 q^{-6} +8 q^{-5} -12 q^{-4} +q^3+16 q^{-3} -4 q^2-15 q^{-2} +8 q+16 q^{-1} -11 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-8 a^2 z^4+3 z^4+2 a^4 z^2-4 a^2 z^2+2 z^2-a^2+1+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^8 z^4+4 a^7 z^5-2 a^7 z^3+8 a^6 z^6-8 a^6 z^4+3 a^6 z^2+10 a^5 z^7-12 a^5 z^5+4 a^5 z^3+8 a^4 z^8-7 a^4 z^6-3 a^4 z^4+2 a^4 z^2-a^4 z^{-2} +a^4+3 a^3 z^9+9 a^3 z^7-26 a^3 z^5+12 a^3 z^3-a^3 z+2 a^3 z^{-1} +14 a^2 z^8-31 a^2 z^6+z^6 a^{-2} +18 a^2 z^4-2 z^4 a^{-2} -4 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +a^2+3 a z^9+3 a z^7+4 z^7 a^{-1} -20 a z^5-10 z^5 a^{-1} +12 a z^3+6 z^3 a^{-1} -a z+2 a z^{-1} +6 z^8-15 z^6+10 z^4-2 z^2- z^{-2} +1 (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          11
5         3 -3
3        51 4
1       63  -3
-1      105   5
-3     78    1
-5    98     1
-7   59      4
-9  37       -4
-11 15        4
-13 3         -3
-151          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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