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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 L11a395's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,11,21,12 X22,19,9,20 X18,14,19,13 X14,18,15,17 X12,21,13,22 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 7, -8, 4, -3, 8, -7, 6, -5, 9, -6}
A Braid Representative
A Morse Link Presentation L11a395 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w^3-4 u v w^2+4 u v w-2 u v-2 u w^3+5 u w^2-5 u w+2 u-2 v w^3+5 v w^2-5 v w+2 v+2 w^3-4 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial 1-4 q^{-1} +8 q^{-2} -11 q^{-3} +15 q^{-4} -15 q^{-5} +17 q^{-6} -12 q^{-7} +9 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{12} z^{-2} +4 a^{10} z^{-2} +4 a^{10}-6 z^2 a^8-5 a^8 z^{-2} -11 a^8+4 z^4 a^6+9 z^2 a^6+2 a^6 z^{-2} +7 a^6-z^6 a^4-2 z^4 a^4-z^2 a^4+z^4 a^2+z^2 a^2 (db)
Kauffman polynomial z^5 a^{13}-3 z^3 a^{13}+3 z a^{13}-a^{13} z^{-1} +2 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+a^{12} z^{-2} -2 a^{12}+2 z^7 a^{11}+2 z^5 a^{11}-12 z^3 a^{11}+13 z a^{11}-5 a^{11} z^{-1} +2 z^8 a^{10}+4 z^6 a^{10}-13 z^4 a^{10}+14 z^2 a^{10}+4 a^{10} z^{-2} -10 a^{10}+2 z^9 a^9+z^7 a^9+3 z^5 a^9-16 z^3 a^9+21 z a^9-9 a^9 z^{-1} +z^{10} a^8+4 z^8 a^8-4 z^6 a^8-8 z^4 a^8+20 z^2 a^8+5 a^8 z^{-2} -14 a^8+6 z^9 a^7-9 z^7 a^7+4 z^5 a^7-6 z^3 a^7+11 z a^7-5 a^7 z^{-1} +z^{10} a^6+8 z^8 a^6-22 z^6 a^6+11 z^4 a^6+6 z^2 a^6+2 a^6 z^{-2} -7 a^6+4 z^9 a^5-4 z^7 a^5-8 z^5 a^5+6 z^3 a^5+6 z^8 a^4-15 z^6 a^4+8 z^4 a^4-2 z^2 a^4+4 z^7 a^3-10 z^5 a^3+5 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^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 \
1           11
-1          3 -3
-3         51 4
-5        74  -3
-7       84   4
-9      77    0
-11     108     2
-13    510      5
-15   47       -3
-17  15        4
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology

(db, data source)

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