L11a546

L11a545

L11a547

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a546 at Knotilus!

Link Presentations

[edit Notes on L11a546's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,16,11,15 X20,9,21,10 X22,13,19,14 X14,21,15,22 X10,19,5,20 X8,18,9,17 X16,8,17,7 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, 9, -8, 4, -7}, {11, -2, 5, -6, 3, -9, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+3uvw-uvx^{2}+3uvx-3uv+uw^{2}x^{2}-2uw^{2}x+uw^{2}-2uwx^{2}+4uwx-2uw+ux^{2}-2ux+u+vw^{2}x^{2}-2vw^{2}x+vw^{2}-2vwx^{2}+4vwx-2vw+vx^{2}-2vx+v-3w^{2}x^{2}+3w^{2}x-w^{2}+3wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}+{\frac {19}{q^{7/2}}}+q^{5/2}-{\frac {22}{q^{5/2}}}-4q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+9{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z^{-1}-3a^{7}z^{-3}-4a^{7}z-6a^{7}z^{-1}+6a^{5}z^{3}+3a^{5}z^{-3}+12a^{5}z+9a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-a^{3}z^{-3}-10a^{3}z-4a^{3}z^{-1}-az^{5}+az^{3}+z^{3}a^{-1}+2az$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+10a^{9}z^{3}-a^{9}z^{-3}-10a^{9}z+5a^{9}z^{-1}+2a^{8}z^{8}-5a^{8}z^{6}+10a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+3a^{7}z^{9}-5a^{7}z^{7}-7a^{7}z^{5}+22a^{7}z^{3}-3a^{7}z^{-3}-23a^{7}z+12a^{7}z^{-1}+a^{6}z^{10}+11a^{6}z^{8}-40a^{6}z^{6}+31a^{6}z^{4}+10a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+9a^{5}z^{9}-8a^{5}z^{7}-34a^{5}z^{5}+49a^{5}z^{3}-3a^{5}z^{-3}-27a^{5}z+12a^{5}z^{-1}+a^{4}z^{10}+22a^{4}z^{8}-61a^{4}z^{6}+41a^{4}z^{4}+3a^{4}z^{-2}-10a^{4}+6a^{3}z^{9}+12a^{3}z^{7}-57a^{3}z^{5}+53a^{3}z^{3}-a^{3}z^{-3}-20a^{3}z+5a^{3}z^{-1}+13a^{2}z^{8}-17a^{2}z^{6}+2a^{2}z^{4}+z^{4}a^{-2}+2a^{2}z^{2}+14az^{7}-21az^{5}+4z^{5}a^{-1}+15az^{3}-z^{3}a^{-1}-6az+9z^{6}-7z^{4}+2z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   6 1   -5 0                 9 3     6 -2               10 7       -3 -4             12 8         4 -6           11 14           3 -8         7 8             -1 -10       6 13               7 -12     2 5                 -3 -14     6                   6 -16 1 2                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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