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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 L11a48's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X20,16,21,15 X16,20,17,19 X4,21,1,22 X12,6,13,5 X10,4,11,3 X22,12,5,11 X14,9,15,10 X2,14,3,13 X8,17,9,18
Gauss code {1, -10, 7, -5}, {6, -1, 2, -11, 9, -7, 8, -6, 10, -9, 3, -4, 11, -2, 4, -3, 5, -8}
A Braid Representative
A Morse Link Presentation L11a48 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-4 v+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+24 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+4 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^3 z+a z-2 z a^{-1} +3 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +10 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +11 z^8 a^{-4} +a^4 z^6-19 z^6 a^{-4} -2 a^4 z^4+14 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} + a^{-4} +7 z^9 a^{-3} +4 a^3 z^7-10 a^3 z^5-19 z^5 a^{-3} +8 a^3 z^3+23 z^3 a^{-3} -a^3 z-13 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+16 z^8 a^{-2} -11 a^2 z^6-38 z^6 a^{-2} +3 a^2 z^4+26 z^4 a^{-2} +a^2 z^2-6 z^2 a^{-2} +a^2+3 a^{-2} +5 a z^9+12 z^9 a^{-1} -a z^7-14 z^7 a^{-1} -16 a z^5-9 z^5 a^{-1} +14 a z^3+18 z^3 a^{-1} -3 a z-10 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +2 z^{10}+11 z^8-27 z^6+13 z^4-2 z^2+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 \
14           11
12          3 -3
10         71 6
8        83  -5
6       127   5
4      128    -4
2     1112     -1
0    1014      4
-2   59       -4
-4  310        7
-6 15         -4
-8 3          3
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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