L10a155

L10a154

L10a156

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a155 at Knotilus!

Link Presentations

[edit Notes on L10a155's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(uvw^{2}-3uvw+2uv+4uw-2u-2vw^{2}+4vw+2w^{2}-3w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+10q^{2}-12q+16-16q^{-1}+15q^{-2}-11q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+4a^{2}-z^{6}-3z^{4}-7z^{2}-2z^{-2}-6+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+3a^{5}z^{7}+7a^{3}z^{7}+16az^{7}+12z^{7}a^{-1}+a^{6}z^{6}-7a^{4}z^{6}-21a^{2}z^{6}+10z^{6}a^{-2}-3z^{6}-8a^{5}z^{5}-27a^{3}z^{5}-38az^{5}-15z^{5}a^{-1}+4z^{5}a^{-3}-3a^{6}z^{4}+4a^{2}z^{4}-12z^{4}a^{-2}+z^{4}a^{-4}-12z^{4}+7a^{5}z^{3}+23a^{3}z^{3}+17az^{3}+z^{3}a^{-1}+3a^{6}z^{2}+4a^{4}z^{2}+3a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}-2a^{5}z-6a^{3}z+2az+6za^{-1}-a^{6}-2a^{2}-5a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   4 1 -3 5                 6     6 3               6 4     -2 1             10 6       4 -1           8 8         0 -3         7 8           -1 -5       4 8             4 -7     3 7               -4 -9   1 5                 4 -11   2                   -2 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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