L10a150

L10a149

L10a151

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a150 at Knotilus!

Link Presentations

[edit Notes on L10a150's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (w-1) \left(2 v w^2-2 v w+v-w^2+2 w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q^{10}-4 q^9+8 q^8-11 q^7+13 q^6-13 q^5+13 q^4-8 q^3+6 q^2-2 q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-8} +3 a^{-2} -4 a^{-4} + a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db)
Kauffman polynomial	$z^4 a^{-12} +4 z^5 a^{-11} -2 z^3 a^{-11} +8 z^6 a^{-10} -9 z^4 a^{-10} +3 z^2 a^{-10} - a^{-10} +9 z^7 a^{-9} -12 z^5 a^{-9} +5 z^3 a^{-9} -z a^{-9} +5 z^8 a^{-8} +2 z^6 a^{-8} -15 z^4 a^{-8} +11 z^2 a^{-8} -3 a^{-8} +z^9 a^{-7} +13 z^7 a^{-7} -29 z^5 a^{-7} +16 z^3 a^{-7} -3 z a^{-7} +7 z^8 a^{-6} -7 z^6 a^{-6} -13 z^4 a^{-6} +18 z^2 a^{-6} + a^{-6} z^{-2} -7 a^{-6} +z^9 a^{-5} +6 z^7 a^{-5} -18 z^5 a^{-5} +11 z^3 a^{-5} +z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} -12 z^4 a^{-4} +16 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +2 z^7 a^{-3} -5 z^5 a^{-3} +2 z^3 a^{-3} +3 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -4 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} -4 a^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   3   -3 17                 5 1   4 15               6 3     -3 13             7 5       2 11           8 8         0 9         5 5           0 7       3 8             5 5     3 5               -2 3   1 5                 4 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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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