L11a274

L11a273

L11a275

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a274 at Knotilus!

Link Presentations

[edit Notes on L11a274's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(1)^{3}t(2)^{3}-3t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+3t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+4q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-24az^{5}+6z^{5}a^{-1}+12a^{3}z^{3}-33az^{3}+12z^{3}a^{-1}+9a^{3}z-20az+9za^{-1}+2a^{3}z^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-5az^{9}-3z^{9}a^{-1}-2a^{4}z^{8}+a^{2}z^{8}-3z^{8}a^{-2}-2a^{5}z^{7}+7a^{3}z^{7}+25az^{7}+14z^{7}a^{-1}-2z^{7}a^{-3}-2a^{6}z^{6}+4a^{4}z^{6}+6a^{2}z^{6}+12z^{6}a^{-2}-z^{6}a^{-4}+13z^{6}-a^{7}z^{5}+3a^{5}z^{5}-15a^{3}z^{5}-52az^{5}-26z^{5}a^{-1}+7z^{5}a^{-3}+5a^{6}z^{4}-4a^{4}z^{4}-20a^{2}z^{4}-14z^{4}a^{-2}+4z^{4}a^{-4}-29z^{4}+3a^{7}z^{3}+a^{5}z^{3}+16a^{3}z^{3}+48az^{3}+26z^{3}a^{-1}-4z^{3}a^{-3}-2a^{6}z^{2}+3a^{4}z^{2}+14a^{2}z^{2}+8z^{2}a^{-2}-3z^{2}a^{-4}+20z^{2}-2a^{7}z-10a^{3}z-23az-11za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   3 1   -2 4                 3 1     2 2               4 3       -1 0             6 3         3 -2           4 5           1 -4         4 5             -1 -6       2 4               2 -8     2 4                 -2 -10   1 3                   2 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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