L11a150

L11a149

L11a151

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a150 at Knotilus!

Link Presentations

[edit Notes on L11a150's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X4,20,5,19
Gauss code	{1, -4, 3, -11, 7, -8}, {2, -1, 5, -3, 6, -7, 9, -5, 4, -2, 10, -6, 11, -9, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^4-7 u^2 v^3+9 u^2 v^2-5 u^2 v+u^2-3 u v^4+12 u v^3-19 u v^2+12 u v-3 u+v^4-5 v^3+9 v^2-7 v+2}{u v^2}$ (db)
Jones polynomial	$-\frac{20}{q^{9/2}}-q^{7/2}+\frac{27}{q^{7/2}}+5 q^{5/2}-\frac{32}{q^{5/2}}-12 q^{3/2}+\frac{31}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{12}{q^{11/2}}+20 \sqrt{q}-\frac{28}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^5 z^5-a^5 z^3+a^3 z^7+2 a^3 z^5+a^3 z^3+a^3 z^{-1} +a z^7+2 a z^5-z^5 a^{-1} +a z^3-z^3 a^{-1} -a z-a z^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-a^8 z^4+5 a^7 z^7-8 a^7 z^5+3 a^7 z^3+11 a^6 z^8-22 a^6 z^6+13 a^6 z^4-2 a^6 z^2+12 a^5 z^9-19 a^5 z^7+4 a^5 z^5+2 a^5 z^3+5 a^4 z^{10}+16 a^4 z^8-53 a^4 z^6+36 a^4 z^4-6 a^4 z^2+26 a^3 z^9-47 a^3 z^7+18 a^3 z^5+z^5 a^{-3} +a^3 z-a^3 z^{-1} +5 a^2 z^{10}+22 a^2 z^8-59 a^2 z^6+5 z^6 a^{-2} +36 a^2 z^4-3 z^4 a^{-2} -6 a^2 z^2+a^2+14 a z^9-11 a z^7+12 z^7 a^{-1} -9 a z^5-14 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +a z-a z^{-1} +17 z^8-24 z^6+11 z^4-2 z^2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 12 4     -8 0               16 8       8 -2             16 13         -3 -4           16 15           1 -6         12 17             5 -8       8 15               -7 -10     4 12                 8 -12   1 8                   -7 -14   4                     4 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-2$ ${\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{16}$ $r=-1$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16}$ ${\mathbb Z}^{16}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{16}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L11a149 L11a151