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

Visit L11a321 at Knotilus!

Link Presentations

[edit Notes on L11a321's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,11,17,12 X20,5,21,6 X12,4,13,3 X14,8,15,7 X6,14,7,13 X22,17,9,18 X18,21,19,22 X8,9,1,10 X4,19,5,20 X2,16,3,15
Gauss code {1, -11, 4, -10, 3, -6, 5, -9}, {9, -1, 2, -4, 6, -5, 11, -2, 7, -8, 10, -3, 8, -7}
A Braid Representative
A Morse Link Presentation L11a321 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(1) t(2)-t(2)+2) (2 t(2) t(1)-t(1)+1)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{21}{q^{5/2}}-7 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+12 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^5 z^5-2 a^5 z^3-a^5 z+a^3 z^7+3 a^3 z^5+2 a^3 z^3-a^3 z+a z^7+4 a z^5-z^5 a^{-1} +7 a z^3-3 z^3 a^{-1} +5 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+5 a^7 z^3-a^7 z+7 a^6 z^8-16 a^6 z^6+9 a^6 z^4-3 a^6 z^2+6 a^5 z^9-8 a^5 z^7-5 a^5 z^5+5 a^5 z^3-a^5 z+2 a^4 z^{10}+10 a^4 z^8-32 a^4 z^6+24 a^4 z^4-5 a^4 z^2+11 a^3 z^9-19 a^3 z^7+4 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +2 a^2 z^{10}+9 a^2 z^8-25 a^2 z^6+3 z^6 a^{-2} +21 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+2 z^2 a^{-2} +5 a z^9-2 a z^7+5 z^7 a^{-1} -8 a z^5-7 z^5 a^{-1} +15 a z^3+4 z^3 a^{-1} -8 a z-3 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +6 z^8-7 z^6+3 z^4-1 (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          2 -2
4         51 4
2        72  -5
0       115   6
-2      119    -2
-4     109     1
-6    811      3
-8   610       -4
-10  38        5
-12 16         -5
-14 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
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}^{11}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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).

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