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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X22,14,9,13 X20,12,21,11 X12,22,13,21 X18,16,19,15 X8,18,5,17 X16,8,17,7 X14,20,15,19 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-3 - q-2 + 5q-1 - 6 + 12q - 14q2 + 15q3 - 14q4 + 11q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 3q-6 + 7q-4 + 6q-2 + 6 + 8q2 + 2q6 - 2q8 - 3q10 - 4q14 + 2q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 - a-4z-2 - 2a-4 + 2a-4z4 + a-4z6 + 4a-2z-2 + 8a-2 + 6a-2z2 + 3a-2z4 + a-2z6 - 5z-2 - 9 - 7z2 - 2z4 + 2a2z-2 + 3a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 - 6a-8z4 + 4a-8z6 + 4a-7z3 - 13a-7z5 + 7a-7z7 - 3a-6z2 + 10a-6z4 - 14a-6z6 + 7a-6z8 - a-5z-1 + 3a-5z - 5a-5z3 + 5a-5z5 - 5a-5z7 + 4a-5z9 + a-4z-2 - 2a-4 + a-4z2 + 8a-4z4 - 11a-4z6 + 5a-4z8 + a-4z10 - 5a-3z-1 + 13a-3z - 19a-3z3 + 20a-3z5 - 12a-3z7 + 5a-3z9 + 4a-2z-2 - 10a-2 + 15a-2z2 - 17a-2z4 + 8a-2z6 - a-2z8 + a-2z10 - 9a-1z-1 + 21a-1z - 15a-1z3 + a-1z7 + a-1z9 + 5z-2 - 14 + 20z2 - 14z4 + 2z6 + z8 - 5az-1 + 11az - 6az3 - az5 + az7 + 2a2z-2 - 7a2 + 9a2z2 - 5a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 6 3     j = 9               8 5       j = 7             7 6         j = 5           7 8           j = 3         5 7             j = 1       4 10               j = -1     1 2                 j = -3     4                   j = -5 1 1                     j = -7 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, Alternating, 396]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 396]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 14, 9, 13], X[20, 12, 21, 11], > X[12, 22, 13, 21], X[18, 16, 19, 15], X[8, 18, 5, 17], X[16, 8, 17, 7], > X[14, 20, 15, 19], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 8, -7}, > {11, -2, 4, -5, 3, -9, 6, -8, 7, -6, 9, -4, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 -2 5 2 3 4 5 6 7 8 -6 + q - q + - + 12 q - 14 q + 15 q - 14 q + 11 q - 8 q + 4 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 7 6 2 6 8 10 14 16 6 + q + -- + -- + -- + -- + 8 q + 2 q - 2 q - 3 q - 4 q + 2 q - 8 6 4 2 q q q q 18 20 22 24 > q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 396]][a, z]
Out[8]=	2 2 2 2 8 2 5 1 4 2 a 2 z 6 z 2 2 -9 - -- + -- + 3 a - -- - ----- + ----- + ---- - 7 z - -- + ---- + a z - 4 2 2 4 2 2 2 2 6 2 a a z a z a z z a a 4 4 4 6 6 4 z 2 z 3 z z z > 2 z - -- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 396]][a, z]
Out[9]=	2 2 10 2 5 1 4 2 a 1 5 9 5 a -14 - -- - -- - 7 a + -- + ----- + ----- + ---- - ---- - ---- - --- - --- + 4 2 2 4 2 2 2 2 5 3 a z z a a z a z a z z a z a z 2 2 2 3 3 z 13 z 21 z 2 3 z z 15 z 2 2 z > --- + ---- + ---- + 11 a z + 20 z - ---- + -- + ----- + 9 a z - -- + 5 3 a 6 4 2 9 a a a a a a 3 3 3 3 4 4 4 4 z 5 z 19 z 15 z 3 4 6 z 10 z 8 z > ---- - ---- - ----- - ----- - 6 a z - 14 z - ---- + ----- + ---- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 6 17 z 2 4 z 13 z 5 z 20 z 5 6 4 z 14 z > ----- - 5 a z + -- - ----- + ---- + ----- - a z + 2 z + ---- - ----- - 2 9 7 5 3 8 6 a a a a a a a 6 6 7 7 7 7 8 8 11 z 8 z 2 6 7 z 5 z 12 z z 7 8 7 z 5 z > ----- + ---- + a z + ---- - ---- - ----- + -- + a z + z + ---- + ---- - 4 2 7 5 3 a 6 4 a a a a a a a 8 9 9 9 10 10 z 4 z 5 z z z z > -- + ---- + ---- + -- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 4 1 2 4 q 3 10 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 7 q t + 8 q t + 7 q t + 6 q t + 8 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a396

L11a395 L11a397