L10a143

L10a142

L10a144

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a143 at Knotilus!

Link Presentations

[edit Notes on L10a143's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w+2uv^{2}w^{2}-3uv^{2}w+uv^{2}+uvw^{3}-3uvw^{2}+3uvw-uv+uw^{2}-uw+v^{3}w^{2}-v^{3}w+v^{2}w^{3}-3v^{2}w^{2}+3v^{2}w-v^{2}-vw^{3}+3vw^{2}-2vw-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-9q^{-5}+11q^{-6}-11q^{-7}+11q^{-8}-7q^{-9}+5q^{-10}-2q^{-11}+q^{-12}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-3a^{10}z^{2}-2a^{10}z^{-2}-5a^{10}+2a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}+2a^{8}+3a^{6}z^{4}+6a^{6}z^{2}+2a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{14}z^{6}-4a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+2a^{13}z^{7}-6a^{13}z^{5}+4a^{13}z^{3}+a^{13}z+2a^{12}z^{8}-3a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{9}+3a^{11}z^{7}-12a^{11}z^{5}+12a^{11}z^{3}-8a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-13a^{10}z^{6}+14a^{10}z^{4}-16a^{10}z^{2}-2a^{10}z^{-2}+9a^{10}+a^{9}z^{9}+7a^{9}z^{7}-16a^{9}z^{5}+12a^{9}z^{3}-8a^{9}z+2a^{9}z^{-1}+4a^{8}z^{8}-3a^{8}z^{6}+2a^{8}z^{4}-4a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+6a^{7}z^{7}-7a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+6a^{6}z^{6}-9a^{6}z^{4}+7a^{6}z^{2}-2a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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 -3                     1 1 -5                   3 1 -2 -7                 4     4 -9               5 3     -2 -11             6 4       2 -13           6 6         0 -15         5 5           0 -17       2 6             4 -19     3 5               -2 -21   1 4                 3 -23   1                   -1 -25 1                     1

Integral Khovanov Homology (db, data source)

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

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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