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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 L11a348's Link Presentations]

Planar diagram presentation X12,1,13,2 X10,11,1,12 X14,5,15,6 X18,9,19,10 X2,17,3,18 X16,8,17,7 X8493 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code {1, -5, 7, -10, 3, -11, 6, -7, 4, -2}, {2, -1, 9, -3, 8, -6, 5, -4, 10, -8, 11, -9}
A Braid Representative
A Morse Link Presentation L11a348 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(2 t(2) t(1)-t(1)-t(2)+2) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{28}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{26}{q^{7/2}}-\frac{20}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+a^3 z^5+3 a z^5-z^5 a^{-1} -5 a^3 z^3+5 a z^3-2 z^3 a^{-1} +3 a^5 z-8 a^3 z+4 a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -4 a^4 z^{10}-4 a^2 z^{10}-12 a^5 z^9-22 a^3 z^9-10 a z^9-13 a^6 z^8-20 a^4 z^8-18 a^2 z^8-11 z^8-6 a^7 z^7+16 a^5 z^7+36 a^3 z^7+6 a z^7-8 z^7 a^{-1} -a^8 z^6+26 a^6 z^6+59 a^4 z^6+49 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+6 a^5 z^5-2 a^3 z^5+11 a z^5+10 z^5 a^{-1} -z^5 a^{-3} -10 a^6 z^4-33 a^4 z^4-33 a^2 z^4+5 z^4 a^{-2} -5 z^4-6 a^5 z^3-17 a^3 z^3-18 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^6 z^2+a^4 z^2+3 a^2 z^2-2 z^2 a^{-2} -2 z^2-2 a^7 z+3 a^5 z+10 a^3 z+7 a z+2 z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-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 \
8           11
6          3 -3
4         61 5
2        113  -8
0       146   8
-2      1512    -3
-4     1513     2
-6    1115      4
-8   915       -6
-10  512        7
-12 18         -7
-14 5          5
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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