L10a149

L10a148

L10a150

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a149 at Knotilus!

Link Presentations

[edit Notes on L10a149's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,18,15,17 X10,20,11,19 X18,10,19,9 X14,16,5,15 X2,12,3,11
Gauss code	{1, -10, 5, -3}, {9, -4, 6, -8, 7, -6}, {4, -1, 2, -5, 8, -7, 10, -2, 3, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(2t(3)t(2)-t(2)-t(3)+2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-9q^{6}+13q^{5}-15q^{4}+17q^{3}-14q^{2}+12q-6+3q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}+a^{-6}z^{-2}+2a^{-6}+z^{6}a^{-4}+z^{4}a^{-4}-4z^{2}a^{-4}-2a^{-4}z^{-2}-7a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+6z^{2}a^{-2}+a^{-2}z^{-2}+6a^{-2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-6z^{4}a^{-8}+a^{-8}+9z^{7}a^{-7}-15z^{5}a^{-7}+4z^{3}a^{-7}+7z^{8}a^{-6}-5z^{6}a^{-6}-8z^{4}a^{-6}+5z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+2z^{9}a^{-5}+13z^{7}a^{-5}-28z^{5}a^{-5}+9z^{3}a^{-5}+5za^{-5}-2a^{-5}z^{-1}+11z^{8}a^{-4}-12z^{6}a^{-4}-9z^{4}a^{-4}+18z^{2}a^{-4}+2a^{-4}z^{-2}-10a^{-4}+2z^{9}a^{-3}+8z^{7}a^{-3}-16z^{5}a^{-3}+4z^{3}a^{-3}+7za^{-3}-2a^{-3}z^{-1}+4z^{8}a^{-2}+z^{6}a^{-2}-13z^{4}a^{-2}+18z^{2}a^{-2}+a^{-2}z^{-2}-9a^{-2}+4z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+az+3za^{-1}+3z^{6}-6z^{4}+5z^{2}-2$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                     1 -1 15                   4   4 13                 5 1   -4 11               8 4     4 9             9 7       -2 7           8 6         2 5         6 9           3 3       6 8             -2 1     2 8               6 -1   1 4                 -3 -3   2                   2 -5 1                     -1

Integral Khovanov Homology (db, data source)

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

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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