From Knot Atlas
Jump to: navigation, search






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

Visit L11a101 at Knotilus!

Link Presentations

[edit Notes on L11a101's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X20,11,21,12 X8,19,9,20 X18,9,19,10 X10,17,11,18 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X2,16,3,15
Gauss code {1, -11, 9, -7}, {8, -1, 2, -4, 5, -6, 3, -9, 10, -8, 11, -2, 6, -5, 4, -3, 7, -10}
A Braid Representative
A Morse Link Presentation L11a101 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2-v+1\right) \left(2 v^2-v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+13 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+6 a^3 z^3+2 a z^3-2 z^3 a^{-1} -2 a^5 z+3 a^3 z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-9 a^7 z^5+7 a^7 z^3+4 a^6 z^8-9 a^6 z^6+5 a^6 z^4-a^6 z^2+4 a^5 z^9-8 a^5 z^7+9 a^5 z^5-10 a^5 z^3+4 a^5 z+2 a^4 z^{10}+2 a^4 z^8-8 a^4 z^6+7 a^4 z^4-4 a^4 z^2+10 a^3 z^9-25 a^3 z^7+34 a^3 z^5+z^5 a^{-3} -25 a^3 z^3-z^3 a^{-3} +6 a^3 z+2 a^2 z^{10}+7 a^2 z^8-18 a^2 z^6+4 z^6 a^{-2} +14 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+6 a z^9-6 a z^7+8 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +7 z^3 a^{-1} -2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +9 z^8-16 z^6+10 z^4-2 z^2-1 (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          3 -3
4         61 5
2        73  -4
0       116   5
-2      109    -1
-4     99     0
-6    710      3
-8   49       -5
-10  27        5
-12 14         -3
-14 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

Back to the top.