From Knot Atlas
Jump to: navigation, search






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

Visit L10a169 at Knotilus!

Compare L10n107.

Japanese family symbol

Link Presentations

[edit Notes on L10a169's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X14,9,11,10 X20,13,15,14 X10,19,5,20 X18,12,19,11 X4,17,1,18
Gauss code {1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 6, -8}, {4, -5, 10, -9, 8, -7}
A Braid Representative
A Morse Link Presentation L10a169 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1)^2 (x-1)^2}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -q^{7/2}+5 q^{5/2}-11 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-3 a^3 z^3+3 a z^3-z^3 a^{-1} -a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3} (db)
Kauffman polynomial a^7 z^5+5 a^6 z^6-4 a^6 z^4+11 a^5 z^7-17 a^5 z^5+8 a^5 z^3-a^5 z^{-3} +a^5 z^{-1} +11 a^4 z^8-12 a^4 z^6+3 a^4 z^{-2} -2 a^4+4 a^3 z^9+19 a^3 z^7-48 a^3 z^5+z^5 a^{-3} +24 a^3 z^3-3 a^3 z^{-3} +a^3 z+22 a^2 z^8-34 a^2 z^6+5 z^6 a^{-2} +8 a^2 z^4-4 z^4 a^{-2} +6 a^2 z^{-2} -3 a^2+4 a z^9+19 a z^7+11 z^7 a^{-1} -48 a z^5-17 z^5 a^{-1} +24 a z^3+8 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} +a z+ a^{-1} z^{-1} +11 z^8-12 z^6+3 z^{-2} -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 \
8          11
6         4 -4
4        71 6
2       84  -4
0      147   7
-2     1012    2
-4    1210     2
-6   714      7
-8  48       -4
-10 17        6
-12 4         -4
-141          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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.