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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 8q-1 + 12 - 13q + 13q2 - 11q3 + 10q4 - 5q5 + 3q6 - q7

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + q-6 - 3q-4 + 3q-2 + 1 + 2q2 + 4q4 + 2q6 + 6q8 + 2q10 + 3q12 + 4q14 - 2q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4z-2 + 2a-4 + a-4z4 - 2a-2z-2 - 3a-2 + 2a-2z4 + z-2 - 2z2 + z4 - 2a2z2 + a4

Kauffman Polynomial:

4a-7z3 - 4a-7z5 + a-7z7 - 9a-6z2 + 18a-6z4 - 13a-6z6 + 3a-6z8 + 2a-5z3 + 4a-5z5 - 9a-5z7 + 3a-5z9 - a-4z-2 + 8a-4 - 26a-4z2 + 39a-4z4 - 29a-4z6 + 5a-4z8 + a-4z10 + 2a-3z-1 - 11a-3z + 17a-3z3 - 2a-3z5 - 14a-3z7 + 6a-3z9 - 2a-2z-2 + 13a-2 - 29a-2z2 + 39a-2z4 - 30a-2z6 + 7a-2z8 + a-2z10 + 2a-1z-1 - 11a-1z + 23a-1z3 - 16a-1z5 + 3a-1z9 - z-2 + 5 - 10z2 + 15z4 - 11z6 + 5z8 + 2az3 - 4az5 + 4az7 - 2a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

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 = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 7 2     j = 7               6 5       j = 5             7 5         j = 3           6 6           j = 1         6 7             j = -1       3 7               j = -3     2 5                 j = -5     3                   j = -7 1 2                     j = -9 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, 455]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 455]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[18, 8, 19, 7], > X[16, 10, 17, 9], X[8, 18, 9, 17], X[22, 16, 13, 15], X[12, 20, 5, 19], > X[10, 22, 11, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 5, -9, 3, -8}, > {11, -2, 7, -5, 6, -4, 8, -3, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 5 8 2 3 4 5 6 7 12 + q - -- + -- - - - 13 q + 13 q - 11 q + 10 q - 5 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 -6 3 3 2 4 6 8 1 + q + q - q + -- + q - -- + -- + 2 q + 4 q + 2 q + 6 q + 8 4 2 q q q 10 12 14 16 18 20 22 > 2 q + 3 q + 4 q - 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 455]][a, z]
Out[8]=	2 4 4 2 3 4 -2 1 2 2 z 2 2 4 z 2 z -- - -- + a + z + ----- - ----- - 2 z - -- - 2 a z + z + -- + ---- 4 2 4 2 2 2 6 4 2 a a a z a z a a a
In[9]:=	Kauffman[Link[11, Alternating, 455]][a, z]
Out[9]=	8 13 4 -2 1 2 2 2 11 z 11 z 2 5 + -- + -- + a - z - ----- - ----- + ---- + --- - ---- - ---- - 10 z - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 3 3 9 z 26 z 29 z 4 2 4 z 2 z 17 z 23 z 3 > ---- - ----- - ----- - 2 a z + ---- + ---- + ----- + ----- + 2 a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 3 3 4 18 z 39 z 39 z 2 4 4 4 4 z 4 z > 2 a z + 15 z + ----- + ----- + ----- - 2 a z + a z - ---- + ---- - 6 4 2 7 5 a a a a a 5 5 6 6 6 2 z 16 z 5 3 5 6 13 z 29 z 30 z 2 6 > ---- - ----- - 4 a z + 2 a z - 11 z - ----- - ----- - ----- + 3 a z + 3 a 6 4 2 a a a a 7 7 7 8 8 8 9 9 z 9 z 14 z 7 8 3 z 5 z 7 z 3 z 6 z > -- - ---- - ----- + 4 a z + 5 z + ---- + ---- + ---- + ---- + ---- + 7 5 3 6 4 2 5 3 a a a a a a a a 9 10 10 3 z z z > ---- + --- + --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 2 3 2 5 3 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 7 q t + 5 q t + 6 q t + 5 q t + 7 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a455

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