L11a545

L11a544

L11a546

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a545 at Knotilus!

Link Presentations

[edit Notes on L11a545's Link Presentations]

Planar diagram presentation	X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X18,7,19,8 X8,17,9,18 X10,13,5,14 X22,20,17,19 X16,21,11,22 X20,11,21,12 X4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -6, 4, -7}, {10, -3, 7, -2, 11, -9}, {6, -5, 8, -10, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{2}x-2uv^{2}w^{2}-2uv^{2}wx+3uv^{2}w+uv^{2}x-uv^{2}-2uvw^{2}x+2uvw^{2}+5uvwx-5uvw-3uvx+2uv-2uwx+2uw+2ux-u-v^{2}w^{2}x+2v^{2}w^{2}+2v^{2}wx-2v^{2}w+2vw^{2}x-3vw^{2}-5vwx+5vw+2vx-2v-w^{2}x+w^{2}+3wx-2w-2x+1}{{\sqrt {u}}vw{\sqrt {x}}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {25}{q^{7/2}}}+{\frac {20}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {21}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-3z^{3}a^{7}-6za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+3z^{5}a^{5}+8z^{3}a^{5}+10za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}-z^{7}a^{3}-3z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}+z^{5}a+z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-10a^{9}z^{5}+12a^{9}z^{3}-a^{9}z^{-3}-11a^{9}z+5a^{9}z^{-1}+7a^{8}z^{8}-6a^{8}z^{6}-4a^{8}z^{4}+14a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+5a^{7}z^{9}+7a^{7}z^{7}-35a^{7}z^{5}+50a^{7}z^{3}-3a^{7}z^{-3}-33a^{7}z+12a^{7}z^{-1}+2a^{6}z^{10}+13a^{6}z^{8}-25a^{6}z^{6}+26a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+12a^{5}z^{9}-10a^{5}z^{7}-27a^{5}z^{5}+46a^{5}z^{3}-3a^{5}z^{-3}-33a^{5}z+12a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-37a^{4}z^{6}+11a^{4}z^{4}+14a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+7a^{3}z^{9}-6a^{3}z^{7}-13a^{3}z^{5}+14a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}+9a^{2}z^{8}-20a^{2}z^{6}+10a^{2}z^{4}+a^{2}z^{2}+5az^{7}-10az^{5}+4az^{3}+az+z^{6}-z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     4   4 0                   6 1   -5 -2                 10 4     6 -4               12 8       -4 -6             13 8         5 -8           10 14           4 -10         11 11             0 -12       4 12               8 -14     4 9                 -5 -16   1 6                   5 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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