From Knot Atlas
Jump to: navigation, search






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

Visit L11a546 at Knotilus!

Link Presentations

[edit Notes on L11a546's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,16,11,15 X20,9,21,10 X22,13,19,14 X14,21,15,22 X10,19,5,20 X8,18,9,17 X16,8,17,7 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, 9, -8, 4, -7}, {11, -2, 5, -6, 3, -9, 8, -3}
A Braid Representative
A Morse Link Presentation L11a546 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w^2 x-u v w^2+u v w x^2-3 u v w x+3 u v w-u v x^2+3 u v x-3 u v+u w^2 x^2-2 u w^2 x+u w^2-2 u w x^2+4 u w x-2 u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-2 v w x^2+4 v w x-2 v w+v x^2-2 v x+v-3 w^2 x^2+3 w^2 x-w^2+3 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -\frac{18}{q^{9/2}}+\frac{19}{q^{7/2}}+q^{5/2}-\frac{22}{q^{5/2}}-4 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{11}{q^{11/2}}+9 \sqrt{q}-\frac{15}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -4 a^7 z-6 a^7 z^{-1} +6 a^5 z^3+3 a^5 z^{-3} +12 a^5 z+9 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-a^3 z^{-3} -10 a^3 z-4 a^3 z^{-1} -a z^5+a z^3+z^3 a^{-1} +2 a z (db)
Kauffman polynomial a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-5 a^8 z^6+10 a^8 z^2+3 a^8 z^{-2} -10 a^8+3 a^7 z^9-5 a^7 z^7-7 a^7 z^5+22 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^{10}+11 a^6 z^8-40 a^6 z^6+31 a^6 z^4+10 a^6 z^2+6 a^6 z^{-2} -19 a^6+9 a^5 z^9-8 a^5 z^7-34 a^5 z^5+49 a^5 z^3-3 a^5 z^{-3} -27 a^5 z+12 a^5 z^{-1} +a^4 z^{10}+22 a^4 z^8-61 a^4 z^6+41 a^4 z^4+3 a^4 z^{-2} -10 a^4+6 a^3 z^9+12 a^3 z^7-57 a^3 z^5+53 a^3 z^3-a^3 z^{-3} -20 a^3 z+5 a^3 z^{-1} +13 a^2 z^8-17 a^2 z^6+2 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+14 a z^7-21 a z^5+4 z^5 a^{-1} +15 a z^3-z^3 a^{-1} -6 a z+9 z^6-7 z^4+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 \
6           1-1
4          3 3
2         61 -5
0        93  6
-2       107   -3
-4      128    4
-6     1114     3
-8    78      -1
-10   613       7
-12  25        -3
-14  6         6
-1612          -1
-181           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

Back to the top.