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L11n396 L11n398

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,22,18,19 X13,20,14,21 X19,14,20,15 X21,18,22,5 X8,16,9,15 X2536 X4,12,1,11

Gauss Code:

{{1, -10, 2, -11}, {-7, 6, -8, 5}, {10, -1, 3, -9, 4, -2, 11, -3, -6, 7, 9, -4, -5, 8}}

Jones Polynomial:

q-5 - q-4 + 3q-3 - q-2 + 2q-1 + 1 - q + q2 - 2q3 + 2q4 - q5

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 3q-12 + 5q-10 + 6q-8 + 7q-6 + 4q-4 + 4q-2 + 2 - q2 - q4 - 3q6 - q8 - q10 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 - a-2z-2 - 3a-2 - 2a-2z2 + 4z-2 + 11 + 11z2 + 6z4 + z6 - 5a2z-2 - 11a2 - 9a2z2 - 2a2z4 + 2a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 2a-4z6 + a-3z-1 - 2a-3z + 3a-3z3 - 4a-3z5 + a-3z7 - a-2z-2 + 4a-2 - 5a-2z2 + 5a-1z-1 - 17a-1z + 27a-1z3 - 15a-1z5 + 2a-1z7 - 4z-2 + 17 - 38z2 + 46z4 - 22z6 + 3z8 + 9az-1 - 29az + 32az3 - 7az5 - 4az7 + az9 - 5a2z-2 + 22a2 - 49a2z2 + 56a2z4 - 27a2z6 + 4a2z8 + 5a3z-1 - 15a3z + 11a3z3 + 3a3z5 - 5a3z7 + a3z9 - 2a4z-2 + 10a4 - 19a4z2 + 17a4z4 - 7a4z6 + a4z8

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     1   j = 7                   1 1   j = 5               2 2 1     j = 3             1 2 1       j = 1           2 4 2         j = -1         1 2 4           j = -3       1 2               j = -5     2 1 1               j = -7   1 3                   j = -9                         j = -11 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 397]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 397]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[17, 22, 18, 19], X[13, 20, 14, 21], X[19, 14, 20, 15], X[21, 18, 22, 5], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-7, 6, -8, 5}, > {10, -1, 3, -9, 4, -2, 11, -3, -6, 7, 9, -4, -5, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 3 -2 2 2 3 4 5 1 + q - q + -- - q + - - q + q - 2 q + 2 q - q 3 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 5 6 7 4 4 2 4 6 8 10 2 + q + --- + --- + --- + -- + -- + -- + -- - q - q - 3 q - q - q + 14 12 10 8 6 4 2 q q q q q q q 14 16 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 397]][a, z]
Out[8]=	2 4 2 2 3 2 4 4 1 5 a 2 a 2 z 2 z 11 - -- - 11 a + 3 a + -- - ----- - ---- + ---- + 11 z - -- - ---- - 2 2 2 2 2 2 4 2 a z a z z z a a 2 2 4 2 4 2 4 6 > 9 a z + a z + 6 z - 2 a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 397]][a, z]
Out[9]=	2 4 3 4 2 4 4 1 5 a 2 a 1 5 9 a 5 a 17 + -- + 22 a + 10 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 2 2 2 2 2 2 3 a z z z a z a z z z a z 2 2 z 2 z 17 z 3 2 3 z 5 z 2 2 > -- - --- - ---- - 29 a z - 15 a z - 38 z + ---- - ---- - 49 a z - 5 3 a 4 2 a a a a 3 3 3 4 4 2 3 z 3 z 27 z 3 3 3 4 7 z > 19 a z - ---- + ---- + ----- + 32 a z + 11 a z + 46 z - ---- + 5 3 a 4 a a a 5 5 5 6 2 4 4 4 z 4 z 15 z 5 3 5 6 2 z > 56 a z + 17 a z + -- - ---- - ----- - 7 a z + 3 a z - 22 z + ---- - 5 3 a 4 a a a 7 7 2 6 4 6 z 2 z 7 3 7 8 2 8 > 27 a z - 7 a z + -- + ---- - 4 a z - 5 a z + 3 z + 4 a z + 3 a a 4 8 9 3 9 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 3 2 1 1 1 2 - + 4 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 11 6 7 5 7 4 5 4 5 3 3 3 5 2 3 2 q t q t q t q t q t q t q t q t 1 2 2 q 3 5 3 2 5 2 5 3 > ---- + --- + --- + 2 q t + 2 q t + 2 q t + q t + 2 q t + q t + 2 q t t q t 7 3 7 4 9 4 11 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n397

L11n396 L11n398