L11a298

L11a297

L11a299

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a298 at Knotilus!

Link Presentations

[edit Notes on L11a298's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)t(1)^{2}+2t(2)^{2}t(1)-t(2)t(1)+2t(1)+2t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+8q^{3/2}-10{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-2za^{5}+z^{5}a^{3}+2z^{3}a^{3}+2z^{5}a+6z^{3}a+5za+az^{-1}+z^{5}a^{-1}+z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-2a^{5}z^{9}-5a^{3}z^{9}-3az^{9}-a^{6}z^{8}+a^{4}z^{8}-4a^{2}z^{8}-6z^{8}+11a^{5}z^{7}+19a^{3}z^{7}-8z^{7}a^{-1}+6a^{6}z^{6}+11a^{4}z^{6}+23a^{2}z^{6}-8z^{6}a^{-2}+10z^{6}-20a^{5}z^{5}-17a^{3}z^{5}+23az^{5}+14z^{5}a^{-1}-6z^{5}a^{-3}-12a^{6}z^{4}-22a^{4}z^{4}-22a^{2}z^{4}+11z^{4}a^{-2}-3z^{4}a^{-4}+2z^{4}+14a^{5}z^{3}-26az^{3}-6z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+8a^{6}z^{2}+9a^{4}z^{2}+4a^{2}z^{2}-4z^{2}a^{-2}-z^{2}-4a^{5}z+10az+4za^{-1}-2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                       1 1 8                     2   -2 6                   4 1   3 4                 4 2     -2 2               6 4       2 0             7 6         -1 -2           4 4           0 -4         4 7             3 -6       3 4               -1 -8     1 4                 3 -10   1 3                   -2 -12   1                     1 -14 1                       -1

Integral Khovanov Homology (db, data source)

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

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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