L11a292

L11a291

L11a293

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a292 at Knotilus!

Link Presentations

[edit Notes on L11a292's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,13,21,14 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,20,5,19
Gauss code	{1, -2, 3, -11, 9, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 10, -9, 5, -4, 11, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {3}{q^{9/2}}}-6q^{7/2}+{\frac {7}{q^{7/2}}}+10q^{5/2}-{\frac {12}{q^{5/2}}}-14q^{3/2}+{\frac {14}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-3a^{3}z-2za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+4az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-8a^{2}z^{8}+z^{8}a^{-2}-3z^{8}a^{-4}-4z^{8}-9a^{3}z^{7}+2az^{7}+27z^{7}a^{-1}+15z^{7}a^{-3}-z^{7}a^{-5}-6a^{4}z^{6}+11a^{2}z^{6}+10z^{6}a^{-2}+12z^{6}a^{-4}+15z^{6}-3a^{5}z^{5}+16a^{3}z^{5}+14az^{5}-28z^{5}a^{-1}-19z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-2a^{2}z^{4}-10z^{4}a^{-2}-14z^{4}a^{-4}-5z^{4}+2a^{5}z^{3}-16a^{3}z^{3}-17az^{3}+17z^{3}a^{-1}+12z^{3}a^{-3}-4z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-4a^{2}z^{2}+3z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}+6a^{3}z+8az-2za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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 12                       1 1 10                     2   -2 8                   4 1   3 6                 6 2     -4 4               8 4       4 2             8 6         -2 0           9 8           1 -2         7 10             3 -4       5 7               -2 -6     2 7                 5 -8   1 5                   -4 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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