L11n441

L11n440

L11n442

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n441 at Knotilus!

Link Presentations

[edit Notes on L11n441's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X19,13,20,22 X15,21,16,20 X21,17,22,16 X12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, -9, 10, 11, -3, -8, 9, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(4)^{3}-t(1)t(2)t(4)^{3}+t(2)t(4)^{3}-t(1)t(3)t(4)^{3}-t(2)t(3)t(4)^{3}+t(3)t(4)^{3}-t(4)^{3}-2t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-t(2)t(4)^{2}+t(1)t(3)t(4)^{2}+2t(2)t(3)t(4)^{2}-t(3)t(4)^{2}+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(1)t(3)t(4)-2t(2)t(3)t(4)+t(3)t(4)-t(1)+t(1)t(2)-t(2)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+3{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+2a^{9}z^{-3}-4za^{7}-11a^{7}z^{-1}-7a^{7}z^{-3}+6z^{3}a^{5}+20za^{5}+23a^{5}z^{-1}+9a^{5}z^{-3}-3z^{5}a^{3}-13z^{3}a^{3}-22za^{3}-17a^{3}z^{-1}-5a^{3}z^{-3}+3z^{3}a+6za+4az^{-1}+az^{-3}$ (db)
Kauffman polynomial	$a^{9}z^{7}-6a^{9}z^{5}+14a^{9}z^{3}-2a^{9}z^{-3}-16a^{9}z+9a^{9}z^{-1}+a^{8}z^{8}-a^{8}z^{6}-10a^{8}z^{4}+26a^{8}z^{2}+7a^{8}z^{-2}-23a^{8}+a^{7}z^{9}+2a^{7}z^{7}-21a^{7}z^{5}+41a^{7}z^{3}-7a^{7}z^{-3}-39a^{7}z+23a^{7}z^{-1}+6a^{6}z^{8}-10a^{6}z^{6}-29a^{6}z^{4}+73a^{6}z^{2}+19a^{6}z^{-2}-60a^{6}+a^{5}z^{9}+12a^{5}z^{7}-47a^{5}z^{5}+51a^{5}z^{3}-9a^{5}z^{-3}-37a^{5}z+24a^{5}z^{-1}+5a^{4}z^{8}+a^{4}z^{6}-44a^{4}z^{4}+75a^{4}z^{2}+18a^{4}z^{-2}-58a^{4}+11a^{3}z^{7}-29a^{3}z^{5}+27a^{3}z^{3}-5a^{3}z^{-3}-16a^{3}z+12a^{3}z^{-1}+10a^{2}z^{6}-25a^{2}z^{4}+34a^{2}z^{2}+7a^{2}z^{-2}-24a^{2}+3az^{5}+3az^{3}-az^{-3}-2az+2az^{-1}+6z^{2}+z^{-2}-4$ (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 χ 2                   3 -3 0                 3   3 -2               7 4   -3 -4             4 3 1   2 -6           5 7       2 -8         7 4         3 -10       5 11           6 -12     1 1             0 -14     5               5 -16 1 1                 0 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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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