L11a464

L11a463

L11a465

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a464 at Knotilus!

Link Presentations

[edit Notes on L11a464's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,9,21,10 X10,19,11,20 X22,18,15,17 X18,22,19,21 X14,16,5,15 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {10, -4, 8, -9, 7, -6, 9, -8}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-14q^{3}+19q^{2}-20q+21-16q^{-1}+12q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}-a^{4}z^{2}+4a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}-a^{4}+4a^{2}+4a^{-2}-a^{-4}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-z^{4}a^{-6}+5z^{7}a^{-5}+a^{5}z^{5}-10z^{5}a^{-5}-2a^{5}z^{3}+2z^{3}a^{-5}+a^{5}z+za^{-5}+9z^{8}a^{-4}+3a^{4}z^{6}-22z^{6}a^{-4}-6a^{4}z^{4}+11z^{4}a^{-4}+5a^{4}z^{2}+z^{2}a^{-4}-2a^{4}-2a^{-4}+7z^{9}a^{-3}+4a^{3}z^{7}-11z^{7}a^{-3}-3a^{3}z^{5}-2z^{5}a^{-3}-3a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+3za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+11z^{8}a^{-2}+3a^{2}z^{6}-33z^{6}a^{-2}-17a^{2}z^{4}+16z^{4}a^{-2}+18a^{2}z^{2}+10z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-8a^{2}-8a^{-2}+4az^{9}+11z^{9}a^{-1}-20z^{7}a^{-1}-3az^{5}+9z^{5}a^{-1}-az^{3}-z^{3}a^{-1}+4az+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+6z^{8}-10z^{6}-7z^{4}+22z^{2}+2z^{-2}-11$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     4   -4 9                   6 1   5 7                 8 4     -4 5               11 6       5 3             9 8         -1 1           12 11           1 -1         8 13             5 -3       4 8               -4 -5     2 8                 6 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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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