From Knot Atlas
Jump to: navigation, search






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

Visit L11n195 at Knotilus!

Link Presentations

[edit Notes on L11n195's Link Presentations]

Planar diagram presentation X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X9,18,10,19 X19,10,20,11 X22,18,7,17 X14,22,15,21
Gauss code {1, -2, 3, -6, 4, -5}, {5, -1, -8, 9, 2, -3, 7, -11, 6, -4, 10, 8, -9, -7, 11, -10}
A Braid Representative
A Morse Link Presentation L11n195 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-t(1) t(2)^4-2 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial \frac{1}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+3 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-9 z^3 a^5-6 z a^5-a^5 z^{-1} +z^5 a^3+2 z^3 a^3 (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+2 a^{10} z^2+4 a^9 z^7-7 a^9 z^5+2 a^9 z^3-a^9 z+3 a^8 z^8-3 a^8 z^6-a^8 z^4+a^7 z^9+3 a^7 z^7-5 a^7 z^5-a^7 z^3+4 a^7 z-a^7 z^{-1} +4 a^6 z^8-6 a^6 z^6+6 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-a^5 z^7+7 a^5 z^5-10 a^5 z^3+6 a^5 z-a^5 z^{-1} +a^4 z^8+2 a^4 z^4-2 a^4 z^2+4 a^3 z^5-5 a^3 z^3+a^2 z^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 \
0         1-1
-2        3 3
-4       42 -2
-6      52  3
-8     54   -1
-10    55    0
-12   35     2
-14  35      -2
-16 14       3
-18 2        -2
-201         1
Integral Khovanov Homology

(db, data source)

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