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

Visit L11a150 at Knotilus!

Link Presentations

[edit Notes on L11a150's Link Presentations]

Planar diagram presentation X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X4,20,5,19
Gauss code {1, -4, 3, -11, 7, -8}, {2, -1, 5, -3, 6, -7, 9, -5, 4, -2, 10, -6, 11, -9, 8, -10}
A Braid Representative
A Morse Link Presentation L11a150 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-7 u^2 v^3+9 u^2 v^2-5 u^2 v+u^2-3 u v^4+12 u v^3-19 u v^2+12 u v-3 u+v^4-5 v^3+9 v^2-7 v+2}{u v^2} (db)
Jones polynomial -\frac{20}{q^{9/2}}-q^{7/2}+\frac{27}{q^{7/2}}+5 q^{5/2}-\frac{32}{q^{5/2}}-12 q^{3/2}+\frac{31}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{12}{q^{11/2}}+20 \sqrt{q}-\frac{28}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^5 z^5-a^5 z^3+a^3 z^7+2 a^3 z^5+a^3 z^3+a^3 z^{-1} +a z^7+2 a z^5-z^5 a^{-1} +a z^3-z^3 a^{-1} -a z-a z^{-1} (db)
Kauffman polynomial a^8 z^6-a^8 z^4+5 a^7 z^7-8 a^7 z^5+3 a^7 z^3+11 a^6 z^8-22 a^6 z^6+13 a^6 z^4-2 a^6 z^2+12 a^5 z^9-19 a^5 z^7+4 a^5 z^5+2 a^5 z^3+5 a^4 z^{10}+16 a^4 z^8-53 a^4 z^6+36 a^4 z^4-6 a^4 z^2+26 a^3 z^9-47 a^3 z^7+18 a^3 z^5+z^5 a^{-3} +a^3 z-a^3 z^{-1} +5 a^2 z^{10}+22 a^2 z^8-59 a^2 z^6+5 z^6 a^{-2} +36 a^2 z^4-3 z^4 a^{-2} -6 a^2 z^2+a^2+14 a z^9-11 a z^7+12 z^7 a^{-1} -9 a z^5-14 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +a z-a z^{-1} +17 z^8-24 z^6+11 z^4-2 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 \
8           11
6          4 -4
4         81 7
2        124  -8
0       168   8
-2      1613    -3
-4     1615     1
-6    1217      5
-8   815       -7
-10  412        8
-12 18         -7
-14 4          4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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