L11n258

L11n257

L11n259

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n258 at Knotilus!

Link Presentations

[edit Notes on L11n258's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-4uvw^{2}+5uvw-3uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v+3w^{3}-5w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-3q^{-6}+6q^{-5}-10q^{-4}+q^{3}+13q^{-3}-2q^{2}-12q^{-2}+7q+13q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-2a^{6}-z^{4}a^{4}+2z^{2}a^{4}+3a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2a^{2}z^{-2}-3a^{2}-2z^{4}-4z^{2}-z^{-2}-3+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$6a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}+a^{6}z^{2}-a^{6}z^{-2}+a^{6}+7a^{5}z^{7}-17a^{5}z^{5}+32a^{5}z^{3}-27a^{5}z+8a^{5}z^{-1}+5a^{4}z^{8}-4a^{4}z^{6}-2a^{4}z^{4}+a^{4}z^{2}-3a^{4}z^{-2}+5a^{4}+a^{3}z^{9}+14a^{3}z^{7}-43a^{3}z^{5}+51a^{3}z^{3}-34a^{3}z+10a^{3}z^{-1}+8a^{2}z^{8}-13a^{2}z^{6}+z^{6}a^{-2}+a^{2}z^{4}-4z^{4}a^{-2}-2a^{2}z^{2}+6z^{2}a^{-2}-2a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}-4a^{-2}+az^{9}+9az^{7}+2z^{7}a^{-1}-30az^{5}-4z^{5}a^{-1}+25az^{3}-10az+4za^{-1}+2az^{-1}-2a^{-1}z^{-1}+3z^{8}-5z^{6}-z^{4}+4z^{2}+z^{-2}-3$ (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 χ 7                   1 1 5                 2 1 -1 3               5     5 1             4 2     -2 -1           9 5       4 -3         6 7         1 -5       7 6           1 -7     3 6             3 -9   3 7               -4 -11   3                 3 -13 3                   -3

Integral Khovanov Homology (db, data source)

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

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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