L11a349

L11a348

L11a350

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a349 at Knotilus!

Link Presentations

[edit Notes on L11a349's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-3t(2)^{2}t(1)+5t(2)t(1)-3t(1)+t(2)^{2}-3t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+10q^{5/2}-18q^{3/2}+23{\sqrt {q}}-{\frac {28}{\sqrt {q}}}+{\frac {27}{q^{3/2}}}-{\frac {24}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-a^{3}z+za^{-3}+az^{7}+2az^{5}-2z^{5}a^{-1}+az^{3}-3z^{3}a^{-1}-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-10a^{3}z^{9}-18az^{9}-8z^{9}a^{-1}-13a^{4}z^{8}-22a^{2}z^{8}-8z^{8}a^{-2}-17z^{8}-9a^{5}z^{7}+5a^{3}z^{7}+26az^{7}+8z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+21a^{4}z^{6}+58a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+50z^{6}-a^{7}z^{5}+12a^{5}z^{5}+13a^{3}z^{5}+8z^{5}a^{-1}+8z^{5}a^{-3}+4a^{6}z^{4}-17a^{4}z^{4}-46a^{2}z^{4}-11z^{4}a^{-2}+2z^{4}a^{-4}-38z^{4}+a^{7}z^{3}-8a^{5}z^{3}-12a^{3}z^{3}-5az^{3}-8z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+6a^{4}z^{2}+14a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+11z^{2}+2a^{5}z+2a^{3}z+2za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   7 1   -6 4                 11 3     8 2               12 7       -5 0             16 11         5 -2           13 14           1 -4         11 14             -3 -6       7 13               6 -8     3 11                 -8 -10   1 7                   6 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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