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DrawMorseLink

PD Presentation:

X10,1,11,2 X18,13,19,14 X12,4,13,3 X2,17,3,18 X14,7,15,8 X8,9,1,10 X16,12,17,11 X22,16,9,15 X4,22,5,21 X20,6,21,5 X6,20,7,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 15q-5/2 + 19q-3/2 - 22q-1/2 + 21q1/2 - 19q3/2 + 13q5/2 - 8q7/2 + 4q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 2q-12 + 3q-8 - 3q-6 + 5q-4 - q-2 + 2 + 3q2 - 2q4 + 5q6 - 2q8 + q12 - 2q14 + q16

HOMFLY-PT Polynomial:

- 2a-3z3 - a-3z5 - a-1z-1 - a-1z + 2a-1z3 + 3a-1z5 + a-1z7 + az-1 + 2az + 3az3 + 3az5 + az7 - a3z - 2a3z3 - a3z5

Kauffman Polynomial:

- 2a-5z3 + 3a-5z5 - a-5z7 + 3a-4z2 - 14a-4z4 + 14a-4z6 - 4a-4z8 + 6a-3z3 - 20a-3z5 + 20a-3z7 - 6a-3z9 + 8a-2z2 - 31a-2z4 + 28a-2z6 - 2a-2z8 - 3a-2z10 - a-1z-1 + 2a-1z + 6a-1z3 - 30a-1z5 + 41a-1z7 - 15a-1z9 + 1 + 7z2 - 32z4 + 44z6 - 12z8 - 3z10 - az-1 + 4az - 14az3 + 17az5 + 6az7 - 9az9 - 7a2z4 + 21a2z6 - 14a2z8 + 2a3z - 11a3z3 + 20a3z5 - 14a3z7 - 2a4z2 + 7a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     3   j = 8                   5 1   j = 6                 8 3     j = 4               11 5       j = 2             10 8         j = 0           12 11           j = -2         9 12             j = -4       6 10               j = -6     3 9                 j = -8   1 6                   j = -10   3                     j = -12 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 322]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 322]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 13, 19, 14], X[12, 4, 13, 3], X[2, 17, 3, 18], > X[14, 7, 15, 8], X[8, 9, 1, 10], X[16, 12, 17, 11], X[22, 16, 9, 15], > X[4, 22, 5, 21], X[20, 6, 21, 5], X[6, 20, 7, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9, 10, -11, 5, -6}, > {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 11, -10, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 15 19 22 3/2 q - ---- + ---- - ---- + ---- - ------- + 21 Sqrt[q] - 19 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 13 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 3 3 5 -2 2 4 6 8 12 2 - q + --- - --- + -- - -- + -- - q + 3 q - 2 q + 5 q - 2 q + q - 14 12 8 6 4 q q q q q 14 16 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 322]][a, z]
Out[8]=	3 3 5 5 1 a z 3 2 z 2 z 3 3 3 z 3 z -(---) + - - - + 2 a z - a z - ---- + ---- + 3 a z - 2 a z - -- + ---- + a z z a 3 a 3 a a a 7 5 3 5 z 7 > 3 a z - a z + -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 322]][a, z]
Out[9]=	2 2 3 1 a 2 z 3 2 3 z 8 z 4 2 2 z 1 - --- - - + --- + 4 a z + 2 a z + 7 z + ---- + ---- - 2 a z - ---- + a z z a 4 2 5 a a a 3 3 4 4 6 z 6 z 3 3 3 5 3 4 14 z 31 z > ---- + ---- - 14 a z - 11 a z + a z - 32 z - ----- - ----- - 3 a 4 2 a a a 5 5 5 2 4 4 4 6 4 3 z 20 z 30 z 5 3 5 > 7 a z + 7 a z - a z + ---- - ----- - ----- + 17 a z + 20 a z - 5 3 a a a 6 6 7 7 7 5 5 6 14 z 28 z 2 6 4 6 z 20 z 41 z > 4 a z + 44 z + ----- + ----- + 21 a z - 9 a z - -- + ----- + ----- + 4 2 5 3 a a a a a 8 8 9 9 7 3 7 8 4 z 2 z 2 8 6 z 15 z > 6 a z - 14 a z - 12 z - ---- - ---- - 14 a z - ---- - ----- - 4 2 3 a a a a 10 9 10 3 z > 9 a z - 3 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 9 6 10 12 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 9 2 2 2 4 2 4 3 6 3 6 4 > ---- + 11 t + 10 q t + 8 q t + 11 q t + 5 q t + 8 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a322

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