L11a348

L11a347

L11a349

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a348 at Knotilus!

Link Presentations

[edit Notes on L11a348's Link Presentations]

Planar diagram presentation	X12,1,13,2 X10,11,1,12 X14,5,15,6 X18,9,19,10 X2,17,3,18 X16,8,17,7 X8493 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code	{1, -5, 7, -10, 3, -11, 6, -7, 4, -2}, {2, -1, 9, -3, 8, -6, 5, -4, 10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(2t(2)t(1)-t(1)-t(2)+2)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+17{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {28}{q^{3/2}}}-{\frac {30}{q^{5/2}}}+{\frac {26}{q^{7/2}}}-{\frac {20}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-5a^{3}z^{3}+5az^{3}-2z^{3}a^{-1}+3a^{5}z-8a^{3}z+4az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-4a^{4}z^{10}-4a^{2}z^{10}-12a^{5}z^{9}-22a^{3}z^{9}-10az^{9}-13a^{6}z^{8}-20a^{4}z^{8}-18a^{2}z^{8}-11z^{8}-6a^{7}z^{7}+16a^{5}z^{7}+36a^{3}z^{7}+6az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+26a^{6}z^{6}+59a^{4}z^{6}+49a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+6a^{5}z^{5}-2a^{3}z^{5}+11az^{5}+10z^{5}a^{-1}-z^{5}a^{-3}-10a^{6}z^{4}-33a^{4}z^{4}-33a^{2}z^{4}+5z^{4}a^{-2}-5z^{4}-6a^{5}z^{3}-17a^{3}z^{3}-18az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{6}z^{2}+a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}-2a^{7}z+3a^{5}z+10a^{3}z+7az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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 8                       1 1 6                     3   -3 4                   6 1   5 2                 11 3     -8 0               14 6       8 -2             15 12         -3 -4           15 13           2 -6         11 15             4 -8       9 15               -6 -10     5 12                 7 -12   1 8                   -7 -14   5                     5 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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