L11a261

L11a260

L11a262

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a261 at Knotilus!

Link Presentations

[edit Notes on L11a261's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+t(1)t(2)-t(2)+1\right)\left(t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+4q^{5/2}-{\frac {21}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+7a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-3a^{5}z+6a^{3}z-2az-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-5a^{5}z^{9}-11a^{3}z^{9}-6az^{9}-5a^{6}z^{8}-6a^{4}z^{8}-10a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+11a^{5}z^{7}+24a^{3}z^{7}+2az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+13a^{6}z^{6}+22a^{4}z^{6}+25a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}-13a^{5}z^{5}-27a^{3}z^{5}+8az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-12a^{6}z^{4}-25a^{4}z^{4}-20a^{2}z^{4}+5z^{4}a^{-2}-5z^{4}-5a^{7}z^{3}+13a^{5}z^{3}+22a^{3}z^{3}-4az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}+6a^{6}z^{2}+14a^{4}z^{2}+8a^{2}z^{2}-z^{2}a^{-2}+z^{2}-6a^{5}z-11a^{3}z-3az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               11 6       5 -2             11 10         -1 -4           10 9           1 -6         7 11             4 -8       5 10               -5 -10     2 7                 5 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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