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

Visit L11a367 at Knotilus!

Link Presentations

[edit Notes on L11a367's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X18,7,19,8 X8,11,9,12 X20,10,21,9 X22,20,11,19 X10,22,1,21 X4,15,5,16 X6,17,7,18
Gauss code {1, -2, 3, -10, 4, -11, 5, -6, 7, -9}, {6, -1, 2, -3, 10, -4, 11, -5, 8, -7, 9, -8}
A Braid Representative
A Morse Link Presentation L11a367 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^4 v^4-u^4 v^3-u^3 v^4+3 u^3 v^3-u^3 v^2-u^2 v^3+3 u^2 v^2-u^2 v-u v^2+3 u v-u-v+1}{u^2 v^2} (db)
Jones polynomial \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^7 z^7+6 a^7 z^5+11 a^7 z^3+7 a^7 z+a^7 z^{-1} -a^5 z^9-8 a^5 z^7-23 a^5 z^5-29 a^5 z^3-14 a^5 z-a^5 z^{-1} +a^3 z^7+6 a^3 z^5+10 a^3 z^3+4 a^3 z (db)
Kauffman polynomial a^{13} z^3-a^{13} z+2 a^{12} z^4-2 a^{12} z^2+2 a^{11} z^5-a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+2 a^9 z^7-4 a^9 z^5+3 a^9 z^3-a^9 z+2 a^8 z^8-6 a^8 z^6+5 a^8 z^4-2 a^8 z^2+2 a^7 z^9-9 a^7 z^7+15 a^7 z^5-16 a^7 z^3+8 a^7 z-a^7 z^{-1} +a^6 z^{10}-3 a^6 z^8-2 a^6 z^6+9 a^6 z^4-7 a^6 z^2+a^6+4 a^5 z^9-24 a^5 z^7+48 a^5 z^5-42 a^5 z^3+15 a^5 z-a^5 z^{-1} +a^4 z^{10}-4 a^4 z^8+10 a^4 z^4-6 a^4 z^2+2 a^3 z^9-13 a^3 z^7+27 a^3 z^5-21 a^3 z^3+5 a^3 z+a^2 z^8-6 a^2 z^6+10 a^2 z^4-4 a^2 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 \
2           1-1
0          1 1
-2         11 0
-4        31  2
-6       22   0
-8      32    1
-10     22     0
-12    33      0
-14   12       1
-16  23        -1
-18 12         1
-20 1          -1
-221           1
Integral Khovanov Homology

(db, data source)

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