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

Visit L11n233 at Knotilus!

Link Presentations

[edit Notes on L11n233's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X6,13,7,14 X14,7,15,8 X8,15,1,16 X17,22,18,9 X21,16,22,17 X19,4,20,5 X5,20,6,21
Gauss code {1, -2, 3, 10, -11, -5, 6, -7}, {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}
A Braid Representative
A Morse Link Presentation L11n233 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u^3 v^5+u^2 v^4-3 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+2 u v^3-3 u v^2+u v-1}{u^{3/2} v^{5/2}} (db)
Jones polynomial -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{5}{q^{23/2}}+\frac{4}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -2 a^{13} z^3-4 a^{13} z-a^{13} z^{-1} +a^{11} z^7+9 a^{11} z^5+23 a^{11} z^3+19 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-37 a^9 z^3-19 a^9 z-2 a^9 z^{-1} (db)
Kauffman polynomial a^{18} z^4-a^{18} z^2+3 a^{17} z^5-5 a^{17} z^3+a^{17} z+3 a^{16} z^6-4 a^{16} z^4+a^{15} z^7+3 a^{15} z^5-7 a^{15} z^3+3 a^{15} z+3 a^{14} z^6-2 a^{14} z^4-a^{14} z^2+a^{14}+4 a^{13} z^5-6 a^{13} z^3+5 a^{13} z-a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-21 a^{12} z^2+3 a^{12}+a^{11} z^9-10 a^{11} z^7+32 a^{11} z^5-41 a^{11} z^3+22 a^{11} z-3 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+23 a^{10} z^4-19 a^{10} z^2+3 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+19 a^9 z-2 a^9 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
-10           11
-12         1  1
-14       2    2
-16      211   -2
-18     32     1
-20    221     -1
-22   33       0
-24  23        1
-26 12         -1
-28 2          2
-301           -1
Integral Khovanov Homology

(db, data source)

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