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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 L11a373's Link Presentations]

Planar diagram presentation X12,1,13,2 X14,3,15,4 X18,10,19,9 X16,6,17,5 X22,7,11,8 X6,21,7,22 X20,15,21,16 X8,18,9,17 X4,20,5,19 X2,11,3,12 X10,13,1,14
Gauss code {1, -10, 2, -9, 4, -6, 5, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, -3, 9, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L11a373 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^4 v^3-u^4 v^2+u^3 v^4-4 u^3 v^3+6 u^3 v^2-4 u^3 v+u^3-2 u^2 v^4+7 u^2 v^3-11 u^2 v^2+7 u^2 v-2 u^2+u v^4-4 u v^3+6 u v^2-4 u v+u-v^2+v}{u^2 v^2} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^5 z^5-3 a^5 z^3-3 a^5 z+a^3 z^7+4 a^3 z^5+7 a^3 z^3+5 a^3 z+a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +2 a z^3-2 z^3 a^{-1} -2 a z-z a^{-1} -a z^{-1} (db)
Kauffman polynomial -2 a^4 z^{10}-2 a^2 z^{10}-5 a^5 z^9-11 a^3 z^9-6 a z^9-5 a^6 z^8-7 a^4 z^8-11 a^2 z^8-9 z^8-3 a^7 z^7+9 a^5 z^7+21 a^3 z^7+a z^7-8 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+24 a^4 z^6+28 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5-5 a^5 z^5-16 a^3 z^5+11 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-9 a^6 z^4-23 a^4 z^4-22 a^2 z^4+5 z^4 a^{-2} -6 z^4-6 a^7 z^3+2 a^5 z^3+9 a^3 z^3-7 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -2 a^8 z^2+3 a^6 z^2+8 a^4 z^2+6 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z-5 a^3 z-a z+2 z a^{-1} -a^2+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         61 5
2        83  -5
0       116   5
-2      119    -2
-4     1010     0
-6    812      4
-8   59       -4
-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}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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

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See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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