L10a168

L10a167

L10a169

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a168 at Knotilus!

Link Presentations

[edit Notes on L10a168's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(x-1)\left(uvwx-uvw-2uvx-2uwx+uw-ux^{2}+2ux-2vwx+vw-vx^{2}+2vx+2wx+x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {14}{q^{5/2}}}-8q^{3/2}+{\frac {12}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+10{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{5}z^{-3}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}-3a^{3}z^{-3}-4a^{3}z^{-1}-za^{-3}-az^{5}+az^{3}+3az^{-3}+2z^{3}a^{-1}-a^{-1}z^{-3}+5az+7az^{-1}-za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-3a^{4}z^{8}-7a^{2}z^{8}-4z^{8}-3a^{5}z^{7}-10a^{3}z^{7}-13az^{7}-6z^{7}a^{-1}-2a^{6}z^{6}+3a^{2}z^{6}-3z^{6}a^{-2}-2z^{6}-a^{7}z^{5}+3a^{5}z^{5}+27a^{3}z^{5}+37az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{6}z^{4}+8a^{4}z^{4}+21a^{2}z^{4}+4z^{4}a^{-2}+20z^{4}+3a^{7}z^{3}-a^{5}z^{3}-32a^{3}z^{3}-45az^{3}-15z^{3}a^{-1}+2z^{3}a^{-3}-15a^{4}z^{2}-36a^{2}z^{2}-z^{2}a^{-2}-22z^{2}-3a^{7}z+3a^{5}z+24a^{3}z+33az+14za^{-1}-za^{-3}-a^{6}+11a^{4}+24a^{2}+13+a^{7}z^{-1}-3a^{5}z^{-1}-12a^{3}z^{-1}-14az^{-1}-6a^{-1}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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 8                     1 1 6                   3 1 -2 4                 5     5 2               5 3     -2 0             9 5       4 -2           7 9         2 -4         7 5           2 -6       2 7             5 -8     4 7               -3 -10   1 5                 4 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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