L11a312

L11a311

L11a313

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a312 at Knotilus!

Link Presentations

[edit Notes on L11a312's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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