L11a492

L11a491

L11a493

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a492 at Knotilus!

Link Presentations

[edit Notes on L11a492's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X22,16,19,15 X20,9,21,10 X8,19,9,20 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X16,22,17,21 X2,14,3,13
Gauss code	{1, -11, 8, -6}, {5, -4, 10, -3}, {7, -1, 2, -5, 4, -8, 9, -7, 11, -2, 3, -10, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-22q+25-19q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-7z^{4}a^{-2}-3a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-a^{2}-3a^{-2}+z^{8}+5z^{6}+10z^{4}+9z^{2}-2z^{-2}+3$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+11a^{2}z^{8}+16z^{8}a^{-2}+7z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}-8az^{7}-24z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-23a^{2}z^{6}-55z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-66z^{6}+a^{5}z^{5}-12a^{3}z^{5}-5az^{5}-17z^{5}a^{-3}-9z^{5}a^{-5}-5a^{4}z^{4}+24a^{2}z^{4}+60z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+78z^{4}-a^{5}z^{3}+4a^{3}z^{3}+13az^{3}+21z^{3}a^{-1}+19z^{3}a^{-3}+6z^{3}a^{-5}-13a^{2}z^{2}-30z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-36z^{2}-a^{3}z-5az-9za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+6a^{-2}+2a^{-4}+7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 9 3     -6 5               11 6       5 3             11 9         -2 1           14 11           3 -1         9 15             6 -3       6 10               -4 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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