L10a20

L10a19

L10a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a20 at Knotilus!

Link Presentations

[edit Notes on L10a20's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X20,15,5,16 X16,7,17,8 X12,18,13,17 X14,10,15,9 X18,14,19,13 X8,19,9,20 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -8, 6, -2, 10, -5, 7, -6, 3, -4, 5, -7, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+9 q^{7/2}-13 q^{5/2}+15 q^{3/2}-17 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} -2 a^3 z+2 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +6 a z^3-7 z^3 a^{-1} +7 a z-7 z a^{-1} +4 a z^{-1} -4 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^4 a^{-6} +4 z^5 a^{-5} -z^3 a^{-5} +a^4 z^6+9 z^6 a^{-4} -3 a^4 z^4-9 z^4 a^{-4} +3 a^4 z^2+4 z^2 a^{-4} -a^4- a^{-4} +3 a^3 z^7+12 z^7 a^{-3} -8 a^3 z^5-18 z^5 a^{-3} +8 a^3 z^3+11 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +4 a^2 z^8+8 z^8 a^{-2} -6 a^2 z^6-3 z^6 a^{-2} -4 a^2 z^4-14 z^4 a^{-2} +10 a^2 z^2+12 z^2 a^{-2} -4 a^2-4 a^{-2} +2 a z^9+2 z^9 a^{-1} +8 a z^7+17 z^7 a^{-1} -31 a z^5-45 z^5 a^{-1} +29 a z^3+33 z^3 a^{-1} -13 a z-13 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +12 z^8-19 z^6-5 z^4+15 z^2-7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   3   3 8                 6 1   -5 6               7 3     4 4             8 6       -2 2           9 7         2 0         7 10           3 -2       5 7             -2 -4     2 7               5 -6   1 5                 -4 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

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