L10a159

L10a158

L10a160

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a159 at Knotilus!

Link Presentations

[edit Notes on L10a159's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,13,12 X2738 X6,13,1,14
Gauss code	{1, -9, 3, -4, 2, -10}, {9, -1, 7, -6, 8, -5}, {10, -3, 4, -2, 5, -7, 6, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{2}t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)^{2}t(3)+t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+2t(1)t(3)+t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-13}-3q^{-12}+6q^{-11}-8q^{-10}+11q^{-9}-10q^{-8}+10q^{-7}-7q^{-6}+5q^{-5}-2q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$z^{2}a^{12}+a^{12}z^{-2}+2a^{12}-3z^{4}a^{10}-10z^{2}a^{10}-2a^{10}z^{-2}-9a^{10}+2z^{6}a^{8}+9z^{4}a^{8}+13z^{2}a^{8}+a^{8}z^{-2}+7a^{8}+z^{6}a^{6}+4z^{4}a^{6}+4z^{2}a^{6}$ (db)
Kauffman polynomial	$a^{16}z^{4}-a^{16}z^{2}+3a^{15}z^{5}-3a^{15}z^{3}+5a^{14}z^{6}-7a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+5a^{13}z^{7}-6a^{13}z^{5}+3a^{13}z^{3}+3a^{12}z^{8}+a^{12}z^{6}-8a^{12}z^{4}+5a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{9}+6a^{11}z^{7}-17a^{11}z^{5}+17a^{11}z^{3}-9a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-16a^{10}z^{6}+22a^{10}z^{4}-23a^{10}z^{2}-2a^{10}z^{-2}+11a^{10}+a^{9}z^{9}+3a^{9}z^{7}-14a^{9}z^{5}+15a^{9}z^{3}-9a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+18a^{8}z^{4}-18a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+4a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 3     3 -11               4 2     -2 -13             6 3       3 -15           4 4         0 -17         7 6           1 -19       3 6             3 -21     3 5               -2 -23   1 4                 3 -25   2                   -2 -27 1                     1

Integral Khovanov Homology (db, data source)

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

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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