L11n394

L11n393

L11n395

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n394 at Knotilus!

Link Presentations

[edit Notes on L11n394's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, -11, -2, 3, 6, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(w^{2}-w+1\right)\left(uw^{2}-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-6}+4q^{-5}-4q^{-4}+q^{3}+8q^{-3}-3q^{2}-8q^{-2}+5q+8q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+z^{6}+5a^{4}z^{4}-12a^{2}z^{4}+4z^{4}-a^{6}z^{2}+7a^{4}z^{2}-10a^{2}z^{2}+4z^{2}-a^{6}+2a^{4}-3a^{2}+2+a^{6}z^{-2}-2a^{4}z^{-2}+a^{2}z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{3}-a^{7}z+4a^{6}z^{4}-6a^{6}z^{2}+a^{6}z^{-2}-a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+9a^{5}z^{3}-a^{5}z-2a^{5}z^{-1}+4a^{4}z^{8}-19a^{4}z^{6}+37a^{4}z^{4}-25a^{4}z^{2}+2a^{4}z^{-2}+2a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-6a^{3}z^{5}+19a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}+8a^{2}z^{8}-35a^{2}z^{6}+z^{6}a^{-2}+51a^{2}z^{4}-3z^{4}a^{-2}-28a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+5a^{2}+2az^{9}-3az^{7}+3z^{7}a^{-1}-10az^{5}-10z^{5}a^{-1}+17az^{3}+6z^{3}a^{-1}-7az-2za^{-1}+4z^{8}-15z^{6}+15z^{4}-8z^{2}+3$ (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   -2 3               3 1   2 1             3 2     -1 -1         1 6 3       2 -3         5 5         0 -5       4 4           0 -7   1 2 5             4 -9   3 3               0 -11   3                 3 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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