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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 L11a203's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X18,11,19,12 X16,6,17,5 X4,18,5,17 X22,15,7,16 X12,21,13,22 X20,13,21,14 X14,19,15,20
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 10, -11, 8, -6, 7, -5, 11, -10, 9, -8}
A Braid Representative
A Morse Link Presentation L11a203 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-3 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u-2 v^3+3 v^2-3 v+2}{u v^2} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-4 z^3 a^9-4 z a^9-a^9 z^{-1} +z^7 a^7+5 z^5 a^7+9 z^3 a^7+8 z a^7+2 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+3 z^3 a^5-z a^5-z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} (db)
Kauffman polynomial -z^4 a^{14}+2 z^2 a^{14}-2 z^5 a^{13}+3 z^3 a^{13}-3 z^6 a^{12}+5 z^4 a^{12}-3 z^2 a^{12}-4 z^7 a^{11}+10 z^5 a^{11}-12 z^3 a^{11}+2 z a^{11}-4 z^8 a^{10}+12 z^6 a^{10}-18 z^4 a^{10}+9 z^2 a^{10}-2 a^{10}-3 z^9 a^9+9 z^7 a^9-13 z^5 a^9+9 z^3 a^9-4 z a^9+a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+17 z^6 a^8-31 z^4 a^8+25 z^2 a^8-5 a^8-5 z^9 a^7+20 z^7 a^7-31 z^5 a^7+29 z^3 a^7-12 z a^7+2 a^7 z^{-1} -z^{10} a^6+10 z^6 a^6-14 z^4 a^6+10 z^2 a^6-3 a^6-2 z^9 a^5+6 z^7 a^5-z^5 a^5-3 z^3 a^5-z a^5-2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-z^2 a^4+a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-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 \
0           11
-2          1 -1
-4         41 3
-6        32  -1
-8       63   3
-10      54    -1
-12     55     0
-14    45      1
-16   35       -2
-18  14        3
-20 13         -2
-22 1          1
-241           -1
Integral Khovanov Homology

(db, data source)

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

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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