L10a163

L10a162

L10a164

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a163 at Knotilus!

Link Presentations

[edit Notes on L10a163's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)\left(uvw^{2}-uvw+uw+vw-w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-7}-4q^{-6}+8q^{-5}-12q^{-4}+q^{3}+16q^{-3}-4q^{2}-15q^{-2}+8q+16q^{-1}-11$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-8a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-4a^{2}z^{2}+2z^{2}-a^{2}+1+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{8}z^{4}+4a^{7}z^{5}-2a^{7}z^{3}+8a^{6}z^{6}-8a^{6}z^{4}+3a^{6}z^{2}+10a^{5}z^{7}-12a^{5}z^{5}+4a^{5}z^{3}+8a^{4}z^{8}-7a^{4}z^{6}-3a^{4}z^{4}+2a^{4}z^{2}-a^{4}z^{-2}+a^{4}+3a^{3}z^{9}+9a^{3}z^{7}-26a^{3}z^{5}+12a^{3}z^{3}-a^{3}z+2a^{3}z^{-1}+14a^{2}z^{8}-31a^{2}z^{6}+z^{6}a^{-2}+18a^{2}z^{4}-2z^{4}a^{-2}-4a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+3az^{9}+3az^{7}+4z^{7}a^{-1}-20az^{5}-10z^{5}a^{-1}+12az^{3}+6z^{3}a^{-1}-az+2az^{-1}+6z^{8}-15z^{6}+10z^{4}-2z^{2}-z^{-2}+1$ (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 7                     1 1 5                   3   -3 3                 5 1   4 1               6 3     -3 -1             10 5       5 -3           7 8         1 -5         9 8           1 -7       5 9             4 -9     3 7               -4 -11   1 5                 4 -13   3                   -3 -15 1                     1

Integral Khovanov Homology (db, data source)

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

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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