From Knot Atlas
Jump to: navigation, search






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

Visit L10a112 at Knotilus!

Link Presentations

[edit Notes on L10a112's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X14,6,15,5 X18,7,19,8 X16,10,17,9 X10,11,1,12 X6,14,7,13 X4,19,5,20 X20,15,11,16 X2,18,3,17
Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}
A Braid Representative
A Morse Link Presentation L10a112 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^2-u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{4}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+8 q^{5/2}-\frac{14}{q^{5/2}}-13 q^{3/2}+\frac{16}{q^{3/2}}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -2 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -2 a^3 z+5 a z-4 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4+4 a^5 z^5-a^5 z^3+9 a^4 z^6+z^6 a^{-4} -8 a^4 z^4-2 z^4 a^{-4} +3 a^4 z^2+z^2 a^{-4} +13 a^3 z^7+4 z^7 a^{-3} -20 a^3 z^5-10 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -4 a^3 z-2 z a^{-3} +10 a^2 z^8+6 z^8 a^{-2} -10 a^2 z^6-13 z^6 a^{-2} -3 a^2 z^4+6 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +3 a z^9+3 z^9 a^{-1} +16 a z^7+7 z^7 a^{-1} -47 a z^5-33 z^5 a^{-1} +34 a z^3+28 z^3 a^{-1} -10 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +16 z^8-33 z^6+14 z^4-1 (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         3 3
6        51 -4
4       83  5
2      85   -3
0     108    2
-2    810     2
-4   68      -2
-6  38       5
-8 16        -5
-10 3         3
-121          -1
Integral Khovanov Homology

(db, data source)

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