From Knot Atlas
Jump to: navigation, search






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

Visit L11a341 at Knotilus!

Link Presentations

[edit Notes on L11a341's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X8,18,9,17
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 L11a341 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+5 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-10 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^5-9 a^3 z^3-12 a^3 z-6 a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +9 a z^3-2 z^3 a^{-1} +10 a z+5 a z^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-4 a^6 z^8-13 a^4 z^8-19 a^2 z^8-10 z^8-3 a^7 z^7-6 a^5 z^7-6 a^3 z^7-12 a z^7-9 z^7 a^{-1} -a^8 z^6+5 a^6 z^6+27 a^4 z^6+38 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+29 a^5 z^5+50 a^3 z^5+45 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+4 a^6 z^4-10 a^4 z^4-19 a^2 z^4+4 z^4 a^{-2} -4 z^4-8 a^7 z^3-34 a^5 z^3-60 a^3 z^3-47 a z^3-12 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-8 a^6 z^2-6 a^4 z^2-z^2 a^{-2} +4 a^7 z+18 a^5 z+31 a^3 z+24 a z+7 z a^{-1} +a^8+3 a^6+3 a^4+a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -6 a^3 z^{-1} -5 a z^{-1} -2 a^{-1} 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 \
8           11
6          3 -3
4         71 6
2        83  -5
0       127   5
-2      1210    -2
-4     1010     0
-6    812      4
-8   510       -5
-10  28        6
-12 15         -4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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.