L11a257

L11a256

L11a258

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a257 at Knotilus!

Link Presentations

[edit Notes on L11a257's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-4t(1)^{2}t(2)^{4}+4t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-6t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+4t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {18}{q^{9/2}}}-{\frac {19}{q^{7/2}}}+{\frac {16}{q^{5/2}}}+q^{3/2}-{\frac {13}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+3a^{7}z^{3}+2a^{7}z-a^{5}z^{7}-4a^{5}z^{5}-6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-2a^{3}z^{3}-a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-7z^{3}a^{9}+2za^{9}-6z^{8}a^{8}+12z^{6}a^{8}-11z^{4}a^{8}+2z^{2}a^{8}-5z^{9}a^{7}+8z^{7}a^{7}-6z^{5}a^{7}-3z^{3}a^{7}+za^{7}-2z^{10}a^{6}-5z^{8}a^{6}+18z^{6}a^{6}-21z^{4}a^{6}+6z^{2}a^{6}-10z^{9}a^{5}+24z^{7}a^{5}-25z^{5}a^{5}+12z^{3}a^{5}-a^{5}z^{-1}-2z^{10}a^{4}-5z^{8}a^{4}+17z^{6}a^{4}-10z^{4}a^{4}+2z^{2}a^{4}+a^{4}-5z^{9}a^{3}+7z^{7}a^{3}+2z^{5}a^{3}+za^{3}-a^{3}z^{-1}-6z^{8}a^{2}+13z^{6}a^{2}-4z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+10z^{5}a-6z^{3}a-z^{6}+2z^{4}-z^{2}$ (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 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 8 3     5 -4               9 6       -3 -6             10 7         3 -8           8 9           1 -10         8 10             -2 -12       4 9               5 -14     2 7                 -5 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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