L10a158

L10a157

L10a159

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a158 at Knotilus!

Link Presentations

[edit Notes on L10a158's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(u^{2}v^{2}+u^{2}vw-u^{2}v-u^{2}w+uv^{2}w-uv^{2}+uvw^{2}-3uvw+uv-uw^{2}+uw-v^{2}w-vw^{2}+vw+w^{2}\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+9q^{2}-10q+12-10q^{-1}+9q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)-z^{2}a^{-4}+a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}+2z^{4}+z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+4a^{3}z^{7}-az^{7}-z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}+a^{5}z^{5}-8a^{3}z^{5}-2az^{5}-2z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-7a^{4}z^{4}+9a^{2}z^{4}+9z^{4}a^{-2}-7z^{4}a^{-4}+32z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}-3a^{2}z^{2}-3z^{2}a^{-2}+3z^{2}a^{-4}-12z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   2   2 7                 3 1   -2 5               6 2     4 3             5 4       -1 1           7 5         2 -1         5 7           2 -3       4 5             -1 -5     2 6               4 -7   1 3                 -2 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

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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