From Knot Atlas
Jump to: navigation, search






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

Visit L11a204 at Knotilus!

Link Presentations

[edit Notes on L11a204's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X22,20,7,19 X20,12,21,11 X12,22,13,21 X6718 X4,15,5,16
Gauss code {1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a204 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-4 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-7 u v^2+5 u v-u-v^3+4 v^2-4 v+2}{u v^2} (db)
Jones polynomial q^{3/2}-2 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{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+2 z a^7-z^7 a^5-4 z^5 a^5-4 z^3 a^5+z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-10 z a^3-3 a^3 z^{-1} +z^5 a+4 z^3 a+4 z a+a z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-4 z^7 a^9+6 z^5 a^9-4 z^8 a^8+6 z^6 a^8-3 z^4 a^8+2 z^2 a^8-3 z^9 a^7+5 z^7 a^7-7 z^5 a^7+6 z^3 a^7-3 z a^7-z^{10} a^6-3 z^8 a^6+11 z^6 a^6-16 z^4 a^6+6 z^2 a^6-5 z^9 a^5+14 z^7 a^5-17 z^5 a^5+3 z^3 a^5+4 z a^5-2 a^5 z^{-1} -z^{10} a^4-z^8 a^4+6 z^6 a^4-5 z^4 a^4-3 z^2 a^4+3 a^4-2 z^9 a^3+3 z^7 a^3+4 z^5 a^3-12 z^3 a^3+12 z a^3-3 a^3 z^{-1} -2 z^8 a^2+3 z^6 a^2+6 z^4 a^2-9 z^2 a^2+3 a^2-2 z^7 a+7 z^5 a-7 z^3 a+4 z a-a z^{-1} -z^6+4 z^4-4 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 \
4           1-1
2          1 1
0         31 -2
-2        51  4
-4       64   -2
-6      74    3
-8     66     0
-10    67      -1
-12   36       3
-14  36        -3
-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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

Back to the top.