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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X18,6,19,5 X20,12,21,11 X22,16,7,15 X10,18,11,17 X16,22,17,21 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 4q1/2 - 10q3/2 + 16q5/2 - 23q7/2 + 25q9/2 - 26q11/2 + 22q13/2 - 16q15/2 + 10q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 2 + 5q4 - 4q6 + 4q8 + 2q10 - 2q12 + 6q14 - q16 + 6q18 - 3q22 + 3q24 - 6q26 - q28 + 2q30 - q32

HOMFLY-PT Polynomial:

a-9z-1 + a-9z3 - 2a-7z-1 + a-7z3 - a-7z5 - 3a-5z - 5a-5z3 - 3a-5z5 + a-3z-1 + 4a-3z + 2a-3z3 - a-3z5 + a-1z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 - 4a-11z3 + 8a-11z5 - 4a-11z7 - 2a-10 + 7a-10z2 - 14a-10z4 + 18a-10z6 - 8a-10z8 + a-9z-1 - 2a-9z + 4a-9z3 - 5a-9z5 + 13a-9z7 - 8a-9z9 - 5a-8 + 19a-8z2 - 39a-8z4 + 43a-8z6 - 13a-8z8 - 3a-8z10 + 2a-7z-1 - 2a-7z + a-7z3 - 14a-7z5 + 31a-7z7 - 17a-7z9 - 3a-6 + 14a-6z2 - 41a-6z4 + 48a-6z6 - 17a-6z8 - 3a-6z10 + 5a-5z - 18a-5z3 + 14a-5z5 + 5a-5z7 - 9a-5z9 + a-4 + 3a-4z2 - 14a-4z4 + 20a-4z6 - 12a-4z8 - a-3z-1 + 5a-3z - 10a-3z3 + 14a-3z5 - 9a-3z7 + 4a-2z4 - 4a-2z6 + a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     3   j = 18                   7 1   j = 16                 9 3     j = 14               13 7       j = 12             13 9         j = 10           12 13           j = 8         11 13             j = 6       6 13               j = 4     4 10                 j = 2   1 7                   j = 0   3                     j = -2 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, 233]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 233]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[18, 6, 19, 5], X[20, 12, 21, 11], > X[22, 16, 7, 15], X[10, 18, 11, 17], X[16, 22, 17, 21], X[14, 10, 15, 9], > X[4, 20, 5, 19], X[2, 7, 3, 8], X[6, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 8, -6, 4, -2, 11, -8, 5, -7, 6, -3, 9, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 4 Sqrt[q] - 10 q + 16 q - 23 q + 25 q - 26 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 22 q - 16 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 12 14 16 18 22 -2 + q + 5 q - 4 q + 4 q + 2 q - 2 q + 6 q - q + 6 q - 3 q + 24 26 28 30 32 > 3 q - 6 q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 233]][a, z]
Out[8]=	3 3 3 3 3 5 5 5 1 2 1 3 z 4 z z z 5 z 2 z z z 3 z z ---- - ---- + ---- - --- + --- + -- + -- - ---- + ---- + -- - -- - ---- - -- 9 7 3 5 3 9 7 5 3 a 7 5 3 a z a z a z a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 233]][a, z]
Out[9]=	2 2 -2 5 3 -4 1 2 1 2 z 2 z 5 z 5 z z 7 z --- - -- - -- + a + ---- + ---- - ---- - --- - --- + --- + --- - --- + ---- + 10 8 6 9 7 3 9 7 5 3 12 10 a a a a z a z a z a a a a a a 2 2 2 3 3 3 3 3 3 4 19 z 14 z 3 z 4 z 4 z z 18 z 10 z z 2 z > ----- + ----- + ---- - ---- + ---- + -- - ----- - ----- + -- + ---- - 8 6 4 11 9 7 5 3 a 12 a a a a a a a a a 4 4 4 4 4 5 5 5 5 14 z 39 z 41 z 14 z 4 z 8 z 5 z 14 z 14 z > ----- - ----- - ----- - ----- + ---- + ---- - ---- - ----- + ----- + 10 8 6 4 2 11 9 7 5 a a a a a a a a a 5 5 6 6 6 6 6 6 7 7 14 z z z 18 z 43 z 48 z 20 z 4 z 4 z 13 z > ----- - -- - --- + ----- + ----- + ----- + ----- - ---- - ---- + ----- + 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 7 7 7 8 8 8 8 9 9 9 31 z 5 z 9 z 8 z 13 z 17 z 12 z 8 z 17 z 9 z > ----- + ---- - ---- - ---- - ----- - ----- - ----- - ---- - ----- - ---- - 7 5 3 10 8 6 4 9 7 5 a a a a a a a a a a 10 10 3 z 3 z > ----- - ----- 8 6 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 3 q 4 6 6 2 8 2 7 q + 4 q + ----- + - + -- + 10 q t + 6 q t + 13 q t + 11 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 > 13 q t + 12 q t + 13 q t + 13 q t + 9 q t + 13 q t + 14 6 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 9 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a233

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