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

Visit L11a380 at Knotilus!

Link Presentations

[edit Notes on L11a380's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X20,15,21,16 X18,6,19,5 X6,11,7,12 X22,7,11,8 X8,18,9,17 X16,10,17,9 X4,20,5,19 X10,21,1,22
Gauss code {1, -2, 3, -10, 5, -6, 7, -8, 9, -11}, {6, -1, 2, -3, 4, -9, 8, -5, 10, -4, 11, -7}
A Braid Representative
A Morse Link Presentation L11a380 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^4 v^2-u^4 v-u^3 v^4+4 u^3 v^3-5 u^3 v^2+3 u^3 v-u^3+2 u^2 v^4-6 u^2 v^3+7 u^2 v^2-6 u^2 v+2 u^2-u v^4+3 u v^3-5 u v^2+4 u v-u-v^3+v^2}{u^2 v^2} (db)
Jones polynomial -q^{5/2}+3 q^{3/2}-7 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7+2 z^5 a^5+5 z^3 a^5+2 z a^5-z^7 a^3-3 z^5 a^3-2 z^3 a^3+a^3 z^{-1} +2 z^5 a+5 z^3 a+z a-a z^{-1} -z^3 a^{-1} -2 z a^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-6 a^8 z^4+3 a^8 z^2+9 a^7 z^7-15 a^7 z^5+14 a^7 z^3-5 a^7 z+9 a^6 z^8-15 a^6 z^6+9 a^6 z^4-a^6 z^2+6 a^5 z^9-5 a^5 z^7-11 a^5 z^5+14 a^5 z^3-5 a^5 z+2 a^4 z^{10}+8 a^4 z^8-31 a^4 z^6+25 a^4 z^4-7 a^4 z^2+10 a^3 z^9-27 a^3 z^7+19 a^3 z^5-8 a^3 z^3+4 a^3 z-a^3 z^{-1} +2 a^2 z^{10}+2 a^2 z^8-21 a^2 z^6+21 a^2 z^4-6 a^2 z^2+a^2+4 a z^9-12 a z^7+z^7 a^{-1} +8 a z^5-4 z^5 a^{-1} -a z^3+5 z^3 a^{-1} +2 a z-a z^{-1} -2 z a^{-1} +3 z^8-11 z^6+12 z^4-4 z^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 \
6           11
4          2 -2
2         51 4
0        62  -4
-2       95   4
-4      97    -2
-6     98     1
-8    710      3
-10   58       -3
-12  27        5
-14 15         -4
-16 2          2
-181           -1
Integral Khovanov Homology

(db, data source)

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

See/edit the Link_Splice_Base (expert).

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