From Knot Atlas
Jump to: navigation, search






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

Visit L11n238 at Knotilus!

Link Presentations

[edit Notes on L11n238's Link Presentations]

Planar diagram presentation X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X17,8,18,9
Gauss code {1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, {10, -1, -6, 9, -8, 2, -11, 6, -7, 8, -9, 7}
A Braid Representative
A Morse Link Presentation L11n238 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^3-2 u^3 v^2-2 u^2 v^3+6 u^2 v^2-6 u^2 v+2 u^2+2 u v^3-6 u v^2+6 u v-2 u-2 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+5 a z^5-2 z^5 a^{-1} -4 a^3 z^3+11 a z^3-7 z^3 a^{-1} +z^3 a^{-3} +a^5 z-8 a^3 z+12 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -5 a^3 z^{-1} +6 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -5 a^2 z^8-3 z^8 a^{-2} -8 z^8-7 a^3 z^7-14 a z^7-10 z^7 a^{-1} -3 z^7 a^{-3} -3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} -z^6 a^{-4} +12 z^6+17 a^3 z^5+44 a z^5+36 z^5 a^{-1} +9 z^5 a^{-3} +9 z^4 a^{-2} +3 z^4 a^{-4} +6 z^4-6 a^5 z^3-29 a^3 z^3-51 a z^3-37 z^3 a^{-1} -9 z^3 a^{-3} -a^4 z^2-3 a^2 z^2-11 z^2 a^{-2} -3 z^2 a^{-4} -10 z^2+7 a^5 z+21 a^3 z+29 a z+19 z a^{-1} +4 z a^{-3} +a^4+a^2+3 a^{-2} + a^{-4} +3-2 a^5 z^{-1} -5 a^3 z^{-1} -6 a z^{-1} -4 a^{-1} z^{-1} - a^{-3} z^{-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 \
10         1-1
8        2 2
6       41 -3
4      62  4
2     64   -2
0    76    1
-2   67     1
-4  46      -2
-6 26       4
-814        -3
-103         3
Integral Khovanov Homology

(db, data source)

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