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

Visit L11a72 at Knotilus!

Link Presentations

[edit Notes on L11a72's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,21,14 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 11, -2, 3, -9, 8, -4, 6, -5, 7, -3, 9, -8}
A Braid Representative
A Morse Link Presentation L11a72 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 u v^6+3 u v^5-4 u v^4+4 u v^3-3 u v^2+2 u v-u-v^7+2 v^6-3 v^5+4 v^4-4 v^3+3 v^2-2 v}{\sqrt{u} v^{7/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{12}{q^{19/2}}+\frac{11}{q^{21/2}}-\frac{8}{q^{23/2}}+\frac{5}{q^{25/2}}-\frac{2}{q^{27/2}}+\frac{1}{q^{29/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{13} \left(-z^3\right)-4 a^{13} z-3 a^{13} z^{-1} +3 a^{11} z^5+14 a^{11} z^3+19 a^{11} z+7 a^{11} z^{-1} -2 a^9 z^7-11 a^9 z^5-20 a^9 z^3-15 a^9 z-4 a^9 z^{-1} -a^7 z^7-5 a^7 z^5-7 a^7 z^3-2 a^7 z (db)
Kauffman polynomial a^{18} z^4-2 a^{18} z^2+a^{18}+2 a^{17} z^5-2 a^{17} z^3+3 a^{16} z^6-2 a^{16} z^4+4 a^{15} z^7-5 a^{15} z^5+4 a^{15} z^3+4 a^{14} z^8-6 a^{14} z^6+4 a^{14} z^4+a^{14} z^2+3 a^{13} z^9-5 a^{13} z^7+6 a^{13} z^5-11 a^{13} z^3+11 a^{13} z-3 a^{13} z^{-1} +a^{12} z^{10}+4 a^{12} z^8-20 a^{12} z^6+30 a^{12} z^4-24 a^{12} z^2+7 a^{12}+6 a^{11} z^9-24 a^{11} z^7+44 a^{11} z^5-53 a^{11} z^3+30 a^{11} z-7 a^{11} z^{-1} +a^{10} z^{10}+2 a^{10} z^8-19 a^{10} z^6+31 a^{10} z^4-24 a^{10} z^2+7 a^{10}+3 a^9 z^9-14 a^9 z^7+26 a^9 z^5-29 a^9 z^3+17 a^9 z-4 a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+8 a^8 z^4-a^8 z^2+a^7 z^7-5 a^7 z^5+7 a^7 z^3-2 a^7 z (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 \
-6           11
-8          21-1
-10         3  3
-12        42  -2
-14       73   4
-16      55    0
-18     76     1
-20    45      1
-22   47       -3
-24  14        3
-26 14         -3
-28 1          1
-301           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-7 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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