L11a60

L11a59

L11a61

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a60 at Knotilus!

Link Presentations

[edit Notes on L11a60's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{3}-4t(2)^{3}-8t(1)t(2)^{2}+11t(2)^{2}+11t(1)t(2)-8t(2)-4t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {6}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+7q^{5/2}-{\frac {13}{q^{5/2}}}-11q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+z^{3}a^{-3}+a^{3}z-a^{3}z^{-1}-az^{5}-z^{5}a^{-1}+az^{3}-z^{3}a^{-1}+2az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-6az^{9}-4z^{9}a^{-1}-3a^{4}z^{8}-5a^{2}z^{8}-6z^{8}a^{-2}-8z^{8}-3a^{5}z^{7}-4a^{3}z^{7}+8az^{7}+5z^{7}a^{-1}-4z^{7}a^{-3}-2a^{6}z^{6}+a^{4}z^{6}+10a^{2}z^{6}+17z^{6}a^{-2}-z^{6}a^{-4}+25z^{6}-a^{7}z^{5}+3a^{5}z^{5}+12a^{3}z^{5}+5az^{5}+8z^{5}a^{-1}+11z^{5}a^{-3}+3a^{6}z^{4}+a^{4}z^{4}-9a^{2}z^{4}-12z^{4}a^{-2}+2z^{4}a^{-4}-21z^{4}+3a^{7}z^{3}+a^{5}z^{3}-14a^{3}z^{3}-18az^{3}-12z^{3}a^{-1}-6z^{3}a^{-3}+2a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}+6z^{2}-3a^{7}z-2a^{5}z+8a^{3}z+12az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1+a^{7}z^{-1}+a^{5}z^{-1}-a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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                     3   3 6                   4 1   -3 4                 7 3     4 2               7 4       -3 0             9 7         2 -2           8 8           0 -4         5 8             -3 -6       4 8               4 -8     2 5                 -3 -10   1 5                   4 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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