L11a296

L11a295

L11a297

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a296 at Knotilus!

Link Presentations

[edit Notes on L11a296's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X18,8,19,7 X16,6,17,5 X20,13,21,14 X22,15,9,16 X14,21,15,22 X6,18,7,17 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, 5, -7, 6, -4, 8, -3, 9, -5, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}+3t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {11}{q^{7/2}}}+3q^{5/2}-{\frac {14}{q^{5/2}}}-6q^{3/2}+{\frac {14}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+9{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+5a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-4a^{5}z^{3}+10a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}-5a^{5}z+9a^{3}z-2az-2za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-6a^{3}z^{9}-3az^{9}-3a^{6}z^{8}-3a^{4}z^{8}-5a^{2}z^{8}-5z^{8}-2a^{7}z^{7}+10a^{5}z^{7}+19a^{3}z^{7}+2az^{7}-5z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+14a^{4}z^{6}+18a^{2}z^{6}-3z^{6}a^{-2}+11z^{6}+6a^{7}z^{5}-21a^{5}z^{5}-35a^{3}z^{5}+5az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+4a^{8}z^{4}-10a^{6}z^{4}-23a^{4}z^{4}-25a^{2}z^{4}+6z^{4}a^{-2}-10z^{4}-4a^{7}z^{3}+27a^{5}z^{3}+35a^{3}z^{3}-8az^{3}-10z^{3}a^{-1}+2z^{3}a^{-3}-4a^{8}z^{2}+7a^{6}z^{2}+18a^{4}z^{2}+11a^{2}z^{2}-z^{2}a^{-2}+3z^{2}-10a^{5}z-17a^{3}z-3az+4za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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                     2   -2 4                   4 1   3 2                 5 2     -3 0               8 4       4 -2             8 7         -1 -4           6 6           0 -6         5 8             3 -8       4 6               -2 -10     1 5                 4 -12   1 4                   -3 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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