L11a55

L11a54

L11a56

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Link Presentations

[edit Notes on L11a55's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X22,16,5,15 X16,9,17,10 X8,21,9,22 X20,17,21,18 X18,12,19,11 X10,20,11,19 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 8, -2, 11, -3, 4, -5, 7, -8, 9, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(v^4-4 v^3+7 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-9 q^{9/2}+\frac{1}{q^{9/2}}+14 q^{7/2}-\frac{4}{q^{7/2}}-19 q^{5/2}+\frac{8}{q^{5/2}}+22 q^{3/2}-\frac{14}{q^{3/2}}-q^{13/2}+4 q^{11/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-9 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -a^3 z+5 a z-9 z a^{-1} +6 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db)
Kauffman polynomial	$-z^{10} a^{-2} -z^{10}-4 a z^9-9 z^9 a^{-1} -5 z^9 a^{-3} -6 a^2 z^8-20 z^8 a^{-2} -9 z^8 a^{-4} -17 z^8-4 a^3 z^7-5 a z^7-z^7 a^{-1} -8 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+12 a^2 z^6+50 z^6 a^{-2} +13 z^6 a^{-4} -4 z^6 a^{-6} +46 z^6+10 a^3 z^5+34 a z^5+45 z^5 a^{-1} +35 z^5 a^{-3} +13 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-5 a^2 z^4-45 z^4 a^{-2} -8 z^4 a^{-4} +5 z^4 a^{-6} -39 z^4-8 a^3 z^3-37 a z^3-58 z^3 a^{-1} -38 z^3 a^{-3} -8 z^3 a^{-5} +z^3 a^{-7} -a^4 z^2+a^2 z^2+17 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +14 z^2+3 a^3 z+15 a z+26 z a^{-1} +18 z a^{-3} +4 z a^{-5} -a^2-3 a^{-2} - a^{-4} -2-2 a z^{-1} -4 a^{-1} z^{-1} -3 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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 14                       1 1 12                     3   -3 10                   6 1   5 8                 8 3     -5 6               11 6       5 4             11 8         -3 2           11 11           0 0         9 13             4 -2       5 9               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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