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

Visit L11a323 at Knotilus!

Link Presentations

[edit Notes on L11a323's Link Presentations]

Planar diagram presentation X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X14,7,15,8 X12,4,13,3 X16,14,17,13 X6,15,7,16 X2,18,3,17
Gauss code {1, -11, 8, -6, 5, -10, 7, -3}, {3, -1, 2, -8, 9, -7, 10, -9, 11, -2, 4, -5, 6, -4}
A Braid Representative
A Morse Link Presentation L11a323 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-3 u v^2+5 u v-3 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+21 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -3 a^3 z+z a^{-3} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-4 z^3 a^{-1} +4 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-7 a^3 z^9-13 a z^9-6 z^9 a^{-1} -10 a^4 z^8-19 a^2 z^8-7 z^8 a^{-2} -16 z^8-8 a^5 z^7-3 a^3 z^7+11 a z^7+2 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+12 a^4 z^6+43 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +42 z^6-a^7 z^5+11 a^5 z^5+23 a^3 z^5+20 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-27 a^2 z^4-8 z^4 a^{-2} +2 z^4 a^{-4} -28 z^4+a^7 z^3-7 a^5 z^3-20 a^3 z^3-23 a z^3-18 z^3 a^{-1} -7 z^3 a^{-3} -2 a^6 z^2-a^4 z^2+5 a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +6 z^2+2 a^5 z+6 a^3 z+8 a z+6 z a^{-1} +2 z a^{-3} +1-a z^{-1} - a^{-1} 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 \
10           1-1
8          3 3
6         61 -5
4        103  7
2       116   -5
0      1410    4
-2     1213     1
-4    1012      -2
-6   612       6
-8  310        -7
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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