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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 L11a113's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X16,8,17,7 X22,12,5,11 X8,22,9,21 X18,9,19,10 X20,17,21,18 X10,19,11,20 X12,16,13,15 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 3, -5, 6, -8, 4, -9, 11, -2, 9, -3, 7, -6, 8, -7, 5, -4}
A Braid Representative
A Morse Link Presentation L11a113 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+8 t(1) t(2)^3-10 t(2)^3-10 t(1) t(2)^2+8 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{13}{q^{9/2}}-q^{7/2}+\frac{16}{q^{7/2}}+4 q^{5/2}-\frac{20}{q^{5/2}}-8 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{7}{q^{11/2}}+13 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -3 z^5 a^3-9 z^3 a^3-10 z a^3-4 a^3 z^{-1} +z^7 a+4 z^5 a+7 z^3 a+4 z a+a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-7 a^3 z^9-4 a z^9-4 a^6 z^8-10 a^4 z^8-13 a^2 z^8-7 z^8-3 a^7 z^7-4 a^5 z^7-a^3 z^7-7 a z^7-7 z^7 a^{-1} -a^8 z^6+5 a^6 z^6+18 a^4 z^6+22 a^2 z^6-4 z^6 a^{-2} +6 z^6+8 a^7 z^5+23 a^5 z^5+26 a^3 z^5+23 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+6 a^6 z^4+3 a^4 z^4-3 a^2 z^4+6 z^4 a^{-2} +3 z^4-8 a^7 z^3-25 a^5 z^3-27 a^3 z^3-17 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-11 a^6 z^2-16 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} -4 z^2+4 a^7 z+15 a^5 z+14 a^3 z+4 a z+z a^{-1} +a^8+4 a^6+7 a^4+4 a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -4 a^3 z^{-1} -a 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         51 4
2        83  -5
0       105   5
-2      119    -2
-4     99     0
-6    711      4
-8   69       -3
-10  28        6
-12 15         -4
-14 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

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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