L10a157

L10a156

L10a158

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a157 at Knotilus!

Link Presentations

[edit Notes on L10a157's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(u^{2}v^{2}w+u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}-uv^{2}w+uv^{2}-uvw^{2}+uvw-uv+uw^{2}-uw-v^{2}-vw+v+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{-6}+3q^{-5}-q^{4}-5q^{-4}+3q^{3}+8q^{-3}-4q^{2}-9q^{-2}+8q+10q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+2a^{2}z^{2}-2z^{2}a^{-2}+z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{3}+3a^{6}z^{4}-a^{6}z^{2}+5a^{5}z^{5}-3a^{5}z^{3}+7a^{4}z^{6}-10a^{4}z^{4}+4a^{4}z^{2}+7a^{3}z^{7}+z^{7}a^{-3}-13a^{3}z^{5}-4z^{5}a^{-3}+5a^{3}z^{3}+4z^{3}a^{-3}+5a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-14z^{6}a^{-2}-a^{2}z^{4}+20z^{4}a^{-2}+3a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+2az^{9}+2z^{9}a^{-1}+az^{7}-5z^{7}a^{-1}-16az^{5}-2z^{5}a^{-1}+11az^{3}+6z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+8z^{8}-30z^{6}+32z^{4}-10z^{2}+2z^{-2}-3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                     1 -1 7                   2   2 5                 2 1   -1 3               6 2     4 1             3 3       0 -1           7 5         2 -3         4 5           1 -5       4 5             -1 -7     2 5               3 -9   1 3                 -2 -11   2                   2 -13 1                     -1

Integral Khovanov Homology (db, data source)

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

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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