L10a134

L10a133

L10a135

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a134 at Knotilus!

Link Presentations

[edit Notes on L10a134's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{2}-2uv^{2}w+uv^{2}-3uvw^{2}+5uvw-2uv+3uw^{2}-3uw+u-v^{2}w^{2}+3v^{2}w-3v^{2}+2vw^{2}-5vw+3v-w^{2}+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+7q^{-4}-4q^{3}-9q^{-3}+8q^{2}+13q^{-2}-11q-14q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+3a^{2}z^{4}+z^{4}a^{-2}+6a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+4a^{2}+a^{-2}-z^{6}-3z^{4}-5z^{2}-2$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-4a^{5}z^{5}+4a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-5a^{4}z^{6}-a^{4}z^{4}+z^{4}a^{-4}+6a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+a^{3}z^{9}+9a^{3}z^{7}-28a^{3}z^{5}+4z^{5}a^{-3}+20a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z^{-1}+8a^{2}z^{8}-9a^{2}z^{6}+8z^{6}a^{-2}-9a^{2}z^{4}-9z^{4}a^{-2}+9a^{2}z^{2}+4z^{2}a^{-2}+a^{2}z^{-2}-3a^{2}-a^{-2}+az^{9}+16az^{7}+9z^{7}a^{-1}-39az^{5}-11z^{5}a^{-1}+27az^{3}+5z^{3}a^{-1}-6az-2za^{-1}+5z^{8}+5z^{6}-22z^{4}+13z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   3   -3 5                 5 1   4 3               6 3     -3 1             8 5       3 -1           8 8         0 -3         5 6           -1 -5       5 9             4 -7     2 4               -2 -9     5                 5 -11 1 2                   -1 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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.

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