L11a315

L11a314

L11a316

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a315 at Knotilus!

Link Presentations

[edit Notes on L11a315's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{4}-2u^{3}v^{3}+u^{3}v^{2}+u^{2}v^{5}-4u^{2}v^{4}+7u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+7uv^{2}-4uv+u+v^{3}-2v^{2}+2v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {3}{q^{27/2}}}+{\frac {6}{q^{25/2}}}-{\frac {11}{q^{23/2}}}+{\frac {14}{q^{21/2}}}-{\frac {15}{q^{19/2}}}+{\frac {15}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-3za^{13}-a^{13}z^{-1}+3z^{5}a^{11}+12z^{3}a^{11}+13za^{11}+3a^{11}z^{-1}-2z^{7}a^{9}-10z^{5}a^{9}-16z^{3}a^{9}-10za^{9}-2a^{9}z^{-1}-z^{7}a^{7}-5z^{5}a^{7}-8z^{3}a^{7}-4za^{7}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-3a^{17}z^{3}+a^{17}z+5a^{16}z^{6}-4a^{16}z^{4}+a^{16}z^{2}+7a^{15}z^{7}-9a^{15}z^{5}+5a^{15}z^{3}+a^{15}z+7a^{14}z^{8}-11a^{14}z^{6}+8a^{14}z^{4}-3a^{14}z^{2}+a^{14}+4a^{13}z^{9}-13a^{13}z^{5}+9a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+a^{12}z^{10}+10a^{12}z^{8}-35a^{12}z^{6}+39a^{12}z^{4}-22a^{12}z^{2}+3a^{12}+7a^{11}z^{9}-19a^{11}z^{7}+20a^{11}z^{5}-24a^{11}z^{3}+16a^{11}z-3a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-26a^{10}z^{6}+32a^{10}z^{4}-17a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-11a^{9}z^{7}+16a^{9}z^{5}-17a^{9}z^{3}+11a^{9}z-2a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+6a^{8}z^{4}+a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   4     4 -12                 5 2     -3 -14               8 4       4 -16             7 5         -2 -18           8 8           0 -20         6 7             1 -22       5 8               -3 -24     2 7                 5 -26   1 4                   -3 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-7$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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